Especially when they don't have dangling private bits like humans do. This seems to be a thing with almost all depictions of humanoid lizard people, that and wristbands like in the photo below. Why is that?
You may not have dangly bits, or bits that you feel you need to cover. You may even be scaly but warm blooded and don't really need to conserve heat. But if you are smart enough to create and carry weapons, you probably have other things you want to carry around, but understand that you can't carry those 4 neat seashells and that human skull you want to drink stuff out of and your club all at the same time. Also, gripping your stuff means you can't have your claws out ready to gut those nasty humans.
So, you have garments/harnesses/belts to help you carry stuff. With pockets to keep stuff from flopping about. Lets use the term 'clothing' very loosely here. It can be a belt, harness, vest, apron, or even stretchy purple Hulk pants. As long as you can attach something to the clothing to carry it, or actually put something in an attached pouch or pocket.
With this, you do not need to carry things in you hands. In addition, heavier things can be carried and the weight of the objects gets distributed across the body according to the design of the clothes. A belt distributes the weight across the hips. A vest or backpack spreads the weight across the shoulders and back. This allows you to carry more weight than one could just by grabbing it and carrying by hand.
Just think about a carpenters tool belt or a soldiers Rucksack to get the idea.
Because exposing your wrist is SHAMEFUL!
You CAD! How dare you despoil our thoughts with your depraved talk of... of...
gasp! Exposed Naked Wrists!? In Public!?
I'm reporting you to the Brood Chamber Matron, Immediately!
Different cultures have different mores.
How much more different must the mores of different species be?
Tribal adult females of the Yanomami routinely walk around in public wearing no clothing.
But if a woman should show herself without her facial sticks, she is shamed and in possible mortal peril from the village elders.
NSFW warning: if you google this tribe, your western sensibilities may get knotted.
How do you know that the lizardpeople in the picture don't have free-hanging equipment?
Lizardfolk don't exist*, they are complete works of fiction. There is no law that states, "If ye includeth lizardfolk in thine media, verily, their genitalia must be reptilian in nature". Every depiction of a lizardperson is how the creator envisioned them, and typically, they're anthropomorphized humans which means they're humans with reptilian attributes (not the other way around).
* If you're uncertain on this point, ask your nearest government representative: they'll be sure to give you an accurate answer on this
For the Same Reasons Humans Do
Lizardfolk, cold blooded or not (that tends to flip between settings) would still want clothing for thermal insulation. I'm going to go for a cold-blooded approach, but this is true for lizardfolk in general. In cold weather furs et al would be a requirement to keep warm, which would be very important in a cold-blooded species where warmth=agility/alertness to a certain extent. It wouldn't be as efficient as it is for us humans, because we radiate heat as a matter of course, but it'd still be better than wandering around naked! I'd imagine winter/arctic lizardmen spending their mornings around the fire largely naked to absorb as much heat as possible, then adding insulation and going about their day.
In a hot environment they would wear flowing garments like Bedouins do to help keep cool, as they have no ability to sweat like humans. Of course if they evolved from desert-dwelling reptiles they may not need quite as much covering as the Bedouins do. If they were out and about at night however I could see them wearing something similar to our arctic lizards, since deserts gets surprisingly chilly.
Anything temperate/seasonal would have the traditional movement from more to less clothes and back again seen everywhere. Though there may be a stylistic choice to wear things that are easily removeable. I'm imagining something like a dress, wherein the top could peel off. Or a jumpsuit that can be unzipped around the waist. That would let your lizardfolk warm up at will without forcing them to either discard or hand-carry whatever clothes they wore when it was colder.
Then of course you can't discount cultural/religious/biological reasons for doing it. Maybe they all wear hats because they have crests that raise/change color when they get aroused/angry/sick/hungry/whatever, and it becomes poor manners to display that to others in public. Maybe a long tail the epitome of sexy, so they wear tail coverings to make their tails look thicker/longer. The tail coverings become more and more elaborate, and the belt that used to hold them on turns into suspenders/shirts/clothing in general as a status symbol. Or maybe it sticks to just tails. Maybe the mid-scales are considered religiously unclean, and thus should be hidden from view. The social possibilities are endless!
That's what I got for main body coverings. Gloves and shoes would probably be a bit less common, as lizard claws/feet are more durable than human ones. But boots in cold/wet weather or over hazardous ground and gloves for very hot/cold object handling would be common enough.
When the high priest says gird your loins and bind your wrists, you do it. Because even if your god or gods don't actually care, the priest believes they do. And she has the power to make your life uncomfortable if you disobey. Not just by making you a societal outcast, but by actually calling down divine fire or plagues of insects.
And that's assuming that the gods don't actually care. If they do care, things could be a lot worse.
Just because you don't have dangly bits, doesn't mean you want sand or insects getting into the bits you do have. Also the concept of armor mentioned by others.
Sure, you don't need the towel strapped around your waist. But all the squishy races seem to. You might not get along with Humans, but your Goblin and Orc trading partners also wear that stuff. And it makes them more comfortable when you wear it, too.
Pants are a built in cushion. Maybe not a full pillow, but they still make it slightly more pleasant sitting on a random rock or stump.
Your people wear clothing. They have done so since before any surviving records. Maybe once, long ago, it was due to an edict of the gods or to make your neighbors more comfortable. But now? Now it is just how things are done.
They don't have Dangly Bits. But they have Pointy Bits.
Lizard bros are more cold-blooded than human bros. So they need to wear more clothes to stay warm. The problem is they cannot wear human-style clothes due to their pointy bits.
It's impossible to design a pull-over garment that won't get torn to shreds. It's slightly easier to design a front-closing garment, but you need to custom make it for the pattern of spikes on each individual. That's expensive and means only that individual can wear it.
Much easier to have several large pieces of fabric that can be worn differently depending on the individual. They also wear bracers, belts and chokers to anchor the garment and stop it flying away.
This style of clothes is awkward to fight in. So when the time comes to fight they throw the fabric away but keep their bracers/chokers/belt. If you zoom out in the picture you will see a big pile of fabric behind each of the lizards.
Because they all would look the same otherwise
All lizardfolk come from a recent evolutive bottleneck (when there were only some dozens breeding pairs in the whole world) and live in very tight familiar clans, so that genetically all of them are very similar* and it is very difficult to tell one from the other.
In the middle of a battle it would be almost impossible for a group of them to understand who is who, and in particular for the chief to give orders to the other tribesmen (tribeslizards), so they have the idea to wear different outfits, in order to quickly recognize every one.
*NB: a small genetical variance doesn't necessarily imply the phenotypes (outer appearances) to be almost identical, but we can say that for the lizardfolk this is the case.
Look at the medieval knights of yore in plate armor. They wore a full set of plate male covering their bodies, while also wearing a set of chain mail. A sheet of metal can protect against a lot of damage, but once pierced, it's practically useless. Having a set of chain mail behind it to catch incoming blades greatly helps.
The scales of the lizard people are probably pretty strong; however if you look closely at their backs and arms, the scales are articulate. When bent at certain angles, various weak points are exposed to attackers. Holding onto a sword or anything probably flares up the scales on the lizard-folk's wrists, so something to keep something sharp from getting in there and disabling them would be wise. They do have claws, but then again, can those claws damage plate armor? If a claw breaks, can it be regrown fast enough to continue the battle?
Take a look at the half orc that the lizard-folk are fighting. There's a good chance that scale mail they're wearing is a full shirt of it underneath the plate armor. The plate provides good protection while also preventing the scales from exposing weak points in the armor.
When fighting someone with a whip or rope, we tend to shield ourselves with our arms. Without wristband covering their scales, the opponent could wrap the weapon around the arm and then yank it really hard; which would bend the scales or rip them out of the arm. The thought of tearing off or bending back one of my fingernails causes me pain, so it would probably hurt a lizard-folk pretty badly too.
Heating is not the only reason we are wearing clothes, for instance, other important function our underwear has is an additional protection against dirt, fungi, bacterial infections.
During Middle Ages adoption of undergarment can be statistically connected with lower rates of specific infections. Whatever imaginary anatomy of lizard people would be, it is safe to assume that fundamental principals of protection of specific body parts can be applicable to them as well.
For the armguards: WAR
Lizards don't use shields, their fighting style just isn't up for it. But they still want to be able to block blows, and their natural scales aren't actually that good, so what do they do? They add a layer of hardened leather on top of their wrists, so they can fight even more aggressively without worrying for loosing their arm.
Some even add spikes onto the armguard, since its a place where few lizardfolk grow them naturally
I have one thing to say. Religion. Now, as far as I know, lizardfolk are somewhat in the early copper age, just figuring out metalworking. So, I imagine they would have some sort of religious customs that require them to wear those wristbands and loincloths, and lizardfolk that have a higher religious standing wear fancier, more high quality wristbands and loincloths.