So, on my post apocalyptic world, most people are relative homebodies compared to today. Nobody travels more than a few days distance, except for soldiers, merchants, and slavers. It’s been 500 years since the apocalypse, and my main characters eventually goes on a legendary quest, to retrieve a (insert Mcguffin here). There is a problem though, and I need your help to get around it. Since nobody travels that often, then local dialects would spring up around different areas, and after 500 years, they would all be mutually unintelligible. So, is there a plausible way to get around this scenario?
One option would be integrating this into your story. Finding a way to communicate with the locals could lead to interesting conflicts and perhaps new plot elements.
But as far as actually communicating…throughout the Middle Ages, educated people throughout the remains of the Roman Empire continued to read and write Latin in the Classical style, imitating Caesar and Cicero, even as the common language around them was evolving into what would eventually become modern French, Italian, Spanish, and so on. Even while the Roman Empire was still around, common people throughout Europe would speak very differently from Cicero's grand prose in their everyday speech, while still understanding the "proper Latin" of the orators perfectly well.
It's likely that there would be significant diglossia in your world, with the most educated people continuing to study the English (or whichever language became dominant) that they found in written texts, while also speaking the common and rapidly diverging local dialects. Your protagonist would be able to communicate with these people through writing at first until they learned enough of the local language to understand it. Possible translators might include scholars and scientists working to preserve pre-apocalypse knowledge, or religious figures keeping their old practices alive.
EDIT: As Futoque pointed out in the comments, trade languages and pidgins (and thus creoles) are also certain to develop wherever there's trade happening. So this will depend on how isolated your people really are. The process is especially fast among related languages: see the Bantu creoles like Lingála that arose in eastern and central Africa.
EDIT AGAIN: Ethan Kaminski has brought up a few other excellent examples. The modern Arabic language has been evolving and diverging since the time of Muħammad. But the Qur'ān, by tradition, remains static and unchanging (since it's considered the direct word of God, and translating or updating it would end up changing its meaning). So written Qur'ānic Arabic remains exactly the same throughout the Muslim world.
Similarly, as Dan Clarke points out, the "dialects" of modern Chinese would normally be considered entirely separate languages (they're only "dialects" for political reasons). But they all use the same writing system, which makes them mutually intelligible in writing, even if they're completely different in speech. As he puts it, "Today you'll often see Chinese people drawing characters on hands and in the air if something isn't understood just by speaking."
Immersion Learning is Fairly Quick
In Afghanistan I and my squad spent 9 months training and conducting combat missions with the Afghan police. Our interpreters were not trustworthy at best and were sometimes even outright traitors. By necessity I was forced to learn the local language, I became fluent in Pashto and Semi Fluent in its sister-dialect Dari in only 6 months. I'm not even all that smart or gifted (otherwise I wouldn't have joined the infantry). It's just that my life and ability to conduct virtually of my duties and tasks literally depended on effective communication. I had no choice but to adapt to the scenario, and found that when you really really need to learn how to yell "NO! DON'T FIRE THE DAMNED RPG IN HERE! WE WILL ALL DIE!" you tend to learn very quickly. Learning dialects is really easy once you've got a functional grasp of the parent language. Pashto was really difficult to learn but once I had it I started picking up on Dari pretty much just by listening carefully to people who were speaking it.
Likewise, half of the Afghan police unit spoke Pashto, half spoke Dari, and none spoke English. So you have a military unit with 3 languages and no reliable interpreters. By the end of it everyone spoke English, and most of us spoke Pashto or Dari, all the Dari guys spoke Pashto, and vice versa. Necessity is a harsh but effective teacher.
Just add religion. Monasteries, preaching thrive and spread in this setup. Their common language can serve as the travel language easily.
To have wide-ranging trade, in anything at all, you'll need a lingua franca that works across large areas because after 500 years it is likely that neighbouring realms speak drastically different dialects let alone the language relationship, none whatsoever, between what people speak hundreds of miles apart.
Depending on your particular apocalyptic scenario it could be much worse as well; if there was mass migration as a result of the collapse event you may have communities that didn't start out speaking related mother tongues cheek by jowl and then those tongues are going to diverge from their origin as things that don't exist anymore are forgotten, new concepts are added, and mass literacy and schooling disappear under direct survival pressure.
In the western world I would expect some version of English to be retained, English is massively adaptable and constantly integrates new words from every culture it encounters whether that's language neighbours or niche groups from within the English speaking world. I would expect a pidgin trade form to be intelligible across vast tracts of the former Americas, Europe and even in parts of Africa, the Middle East and Asia Minor.
This Map gives a hint of the places where some form of English may come out on top as the national language. Also because it is the diplomatic language of the modern era English may also be kept as a cross border language all over the planet, not spoken at home but used to discuss matters of trade and borders with the guys from across the river that you can't otherwise talk to.
TL;DR: starting from countries with a nearly homogeneous spoken language which were suddenly dissolved, in absence of significant mass-migrations, I posit that the language spoken by people scattered around the country would not change up to a point of not being mutually unintelligible within a time-span of 500 years only.
I don't think you need to explain anything.
Take a look at Latin and the fall of the Roman Empire.
Even though there was no point in time in which everyone spoke exactly the same language everywhere, as the vulgar Latin that was actually spoken could differ from region to region, mutual intelligibility is still very strong across several, geographically distant, regions of Europe despite nearly 1600 years have passed since when it was disbanded.
In linguistics, mutual intelligibility is a relationship between languages or dialects in which speakers of different but related varieties can readily understand each other without prior familiarity or special effort.
The following table shows lexical similarity of some European languages, an index factor that is often used as indication of mutual intelligibility of two languages: (source: wikipedia)
The following chart (taken from here) presents the same data in a visually more clear way. Latin-derived languages, also known as Romance languages, are grouped together in the lower-left corner:
At this point it is worth mentioning a very interesting fact: the living language that is closest to ancient (vulgar) Latin is --according to several sources-- Sardinian, the native language of the Sardinia island. And why could that be? Well, historically, Sardinia is one of the few places who have remained the most isolated from mass immigration of new populations following the fall of the Roman Empire.
And this leads me to my main point: of course language drift naturally occurs within a country, being it isolated or not, over the centuries. But it is normally a very slow and mostly insignificant process. Any modern-day Italian can read and understand (with ease) texts written in Archaic Italian over 800 years ago, nearly 600 years before this country even existed. Vulgar Latin, as it is presented on Wikipedia, also sounds reasonably easy to understand to me as an Italian even though I admit it is distinctively different than my own language. I suspect a similar story holds for other languages, i.e. modern English people can probably understand Shakespeare original work of 600 years ago almost fairly well.
Where am I trying to get at?
The main driving force behind language evolution is not drift, but immigration, conquest and/or population "replacement".
It is when people of different languages meet and mix with one another, that the language evolves more rapidly by absorbing and mixing linguistic elements of the people which are being culturally blended into one another. (note: I am speaking of mass-migrations of the period following the fall of the Roman Empire and those with similar dynamics that happened elsewhere; modern-day migrations have a different dynamic and are more easily absorbed into the existing culture)
Since it is not trivial for me to demonstrate this using European maps depicting the barbaric invasions following the fall of the Roman Empire and those of the Barbaric Kingdoms that spurred after that (because mass migrations and population movements were far from being over in several geographical areas), I will try to prove my point by pointing out a single historical example related to Britannia.
The following text is an extract of a brief language-historical analysis taken from here, which shows how the Old English language was affected by the invasion of French-speaking Normans in 1066, and how that dramatically changed it:
Take, for example, this passage from the most famous of all Old English works, Beowulf:
Hwät! we Gâr-Dena in geâr-dagum
þeód-cyninga þrym gefrunon,
hû þâ äðelingas ellen fremedon.
Oft Scyld Scêfing sceaðena þreátum.
Lo! the Spear-Danes' glory through splendid achievements
The folk-kings' former fame we have heard of,
How princes displayed then their prowess-in-battle.
Oft Scyld the Scefing from scathers in numbers...
Old English was spoken and written in Britain from the 5th century to the middle of the 11th century and is really closer to the Germanic mother tongue of the Anglo-Saxons.
With the arrival of the French-speaking Normans in 1066, Old English underwent dramatic changes and by 1350 it had evolved into Middle English. Middle English is easier but still looks like a foreign language much of the time. Here is an example from Chaucer's Canterbury Tales, the most famous work in Middle English:
Ye seken lond and see for your wynnynges,
As wise folk ye knowen all th'estaat
Of regnes; ye been fadres of tydynges
And tales, bothe of pees and of debaat. (The Man of Law's Tale)
You seek land and sea for your winnings,
As wise folk you know all the estate
Of kingdoms; you be fathers of tidings,
And tales, both of peace and of debate.
Ethnic / religious minorities.
If you have a minority population which has retained its individuality since the apocalypse, they might have their own language (and possibly religion) to go with it. For the Jews, Hebrew is the example - two jews from anywhere in the world could communicate using Hebrew. The Coptic christians are another example - a minority population with their own language though not so far flung.
A good parallel for a story and one that has not been done is the gypsies. Gypsies are and have been scattered thru Europe and the Americas, and with their distinct cultural identity they have their own language.
So for your story: you have a far flung ethnic / religious minority with its own religion and language. Maybe they are more or less inbred. Certainly they can be discriminated against and held in contempt (and possibly some fear - my grandma was told gypsies would steal kids for unknown purposes); this would add energy to a story where the travelers are dependent on this individual. Your individual knows where to find others of her kind when they travel.
You could always have him find a universal translator in the rubble.
More to the point though, remember that in societies were dozens of difrent languages and dialects spring up, multilingualism also inevitably increases as well. It would not be unreasonable to have your character simply know multiple languages in this setting.
Even when he goes very far outside his original area, most people will not be surprised to encounter someone who does not speak their language, and may be more willing to try to bridge the gap. Provided they think he's worth their time of course. Or, has just smashed enough heads in if its a mad max world.
in the early middle ages troubadours originating from southern France who, whilst not sharing a common language, went on to have an enormous direct influence on Italian culture (at the time consisting of numerous states all speaking their own dialects) and indirect one on European culture.
The technique they used was to pepper their songs with local dialect equivalents sharing the same meaning as their own Occitan word, giving people enough key words that they could follow the lyrics, combined with gestures and mime. My own personal theory is that Italian hand gestures have their roots in these performers.
Just a recent example I saw that may be relevant to this, in that the spoken form of the language was noticeably different after about 130 years:
"The owner looked at him open-mouthed. He understood what Ahmed was saying, but some of the words he was using were unfamiliar and old-fashioned, and others he didn't understand at all. It was as though Ahmed had arrived not just from Syria, but from another age."
Sounds like your hero needs an interpreter. You say that slavers travel long distances—what about slaves? Could the heroes themselves be itinerant? Merchants, refugees, migrants?
It also seems likely that the written language would have diverged less than the spoken language, so maybe he can fall back to the literary language, with its language and vocabulary remembered from such timeless masterpieces as Twilight and Ready Player One, if he can just get people to understand his accent. Or even communicate in writing.
There could be a common, simplified pidgin language across the region, like the historical lingua franca around the Mediterranean.
There might be a group of reactionaries around who insist on keeping the language and culture “pure.” They’re probably villains to the audience—but, when the heroes realize that they can actually talk to people in far-away places, they have to admit, it did have a useful purpose.
Cultural artifacts freezing language
In a similar way how a written language may become "frozen" during its decline, as Latin has; a spoken language may become "frozen" if there's a wide store of examples of "proper" language that doesn't change.
Assuming that the postapocalyptic peoples have a wide store of entertainment in the original language - songs, audiobooks and movies; where they have jury-rigged a means to play and possibly copy them, but no means (or limited means) of creating new content, then it may well be that each generation effectively re-learns the same old language from old media.