I'm not talking about a single alien civilization that is stranded on earth, I'm talking about multiple civilizations that will cause major overpopulation on earth. So this means that the civilizations have to be able to live on earth-like planets.
My first thought is that their fuel source for some reason decays or doesn't work in the conditions of our solar system or the milky way, how realistic could this be?
Forgotten past - In Sleeping Giants by Sylvain Neuvel an alien race was left on earth as "protectors" but simply forgot their role and interbreed.
Fuel - Like you said the typical Star Trek story line "I need more dilithium crystal"
Lost in time - If a race traveled back in time and the time machine broke and doesn't have the components to fix Back to the Future 3.
Nowhere to go - Think Doctor Who or Men in Black. Some aliens are on earth not because they are stuck but because they have nowhere else to go.
Banishment - Maybe they were bad (intergalactic war) and if they leave the primitive prison of Earth they will be put to death by the galactic federation. Lilo And Stitch :)
Money - Think Dune - They can't afford the stars. So they must patiently build their empire until they can.
Stupidity/Ignorance - In Hitchhikers Guide to the Galaxy (book 2 I think) A planet ejected its "non-essential" personnel in arcs that landed on earth. The "non-essentials" were all bureaucrats and pencil pushers who couldn't build a space ship to save their lives, which of course meant they died. Just not knowing how also works though.
When I read stranded, I think of someone who was traveling and is now forced to suspend that travel unexpectedly. If I decide to emigrate, I'm not stranded. If my flight is late and I have to take the next one, I would use the word stranded only as a bit of bleak humour assuming that the next flight onward is a credible prospect. I'd be delayed, not stranded. So how to strand a civilization?
Now imagine the maps of these space travelers are out of date. The advance party arrives on Earth with plans to grow food, mine metals, build orbital spacedocks, etc. Except that, oops, Earth is inhabited by those pesky humans. The aliens will not (or cannot) proceed with their plans against the objections of the current occupants of Earth.
Give a look at our Earth, and see why population from a given region flee to another one, seeking refuge and becoming refugees:
Extend those on a planetary scale, and you see a Mayflower reaching Earth...
Take a world similar to ours, except thousands of years into the future. We have polluted and consumed all of our resources so we set course to another planet. We already know this planet has consumable water and oxygen. When we arrive we find out we are not the only human like beings traveling to/already existing on this planet. We would then be stranded on a shared planet.
Wormholes
This method of fast than light travel don't requires much advanced technology, the ships that travel around the portals don't need to be much more advanced than our current technology. You could have aliens fleeing their worlds through a natural occurring wormhole and have the thing close behind them or become unstable.
After that, they would have little choice but make landfall on Earth.
Enough individuals to cause major overpopulation on earth is a tall order. That is a lot of spaceships showing up. And you want more than one additional civilizations in the mix.
A way you could do it is merge dimensions. Different versions of Earth are shuffled into one. Like shuffling different decks of cards together. You would need to invent your rules for what stays and what goes. The upshot of this could be multiple Earth civilizations of varying "alienness" are suddenly together in one place which itself may have taken on attributes of its extradimensional counterparts. A cool thing about this shuffle is you do not need to have just 3 or 4 civilizations - you could have a bell curve, with a few major components, then decreasing amounts of increasingly distant dimensional components. Some Earths might be represented by just a handful of individuals.
DC comics has a multiverse that they have shuffled, though not as dramatically as what you propose I do not think. I thought this was a cool image of the different Earths each with its own Flash.
Star Trek Next Generation did a fine job in the Parallels episode with multiple dimensions suddenly shuffled into one. It is too bad they made all the Enterprises look about the same, though.
In Larry Niven's The Fleet of Worlds, he proposes that the core of our galaxy is exploding, causing a space-faring species which evolved closer to the center to race outward in avoidance of a soon-to-be galaxy spanning shock/radiation wave. Although the nature of their chosen space craft (their entire solar system) preempts their having to visit any other planets, I could imagine that other species from that same neighborhood might not be so lucky. Using smaller and hopefully FTL ships those species might jump a few centuries ahead of the wave, and find themselves in proxity to our beautiful little planet. In such a case, many of the travelers might make a pit stop on Earth to...
refuel, rest, repopulate-their-ships, share-tech-with-other-refugees, locate the next pit stop planet, and enjoy Earth's natural beauty and human entertainment.
If during this time, we primitive little primates confiscated their star ships and forced them to teach us how to make some for ourselves then all of the refugees would be stranded on Earth until we catch up with their tech and can join them in their escape.
