I have to say, I'm pleasantly surprised. Normally a question like this is basically a invitation to have a edition-based flame war. But everything everyone's said here so far is both reasonable and correct.
CaptainHellrazor:
[...] 3.5e and 5e seem to be the most popular versions of the game [...] what seems to make them more popular?
I'll answer this specifically. D&D 3.0 (and then 3.5) had a thing they call the Open Gaming License. Which basically means that most of the game's mechanics and material was open for anyone to use without legal or financial penalty. That meant anyone could publish a gaming book using the same system and the same terms. And lots and lots of people did just that.
This made the "D20 System" the sort of lingua franca of role-playing in a way it had never been before, so much so that lots of other games and settings (Star Wars, L5R) published D20 versions of their own games. And Paizo was able to make Pathfinder (which is basically 3.5-revised). And their Pathfinder Society Organized Play was very well conceived and executed, which brought more people into the hobby and kept them playing longer.
5E had a bit of luck and a bit of foresight. The foresight is that the barrier to entry for 5E is the lowest of all the editions. It's very easy to learn and pick-up and (even more importantly) very easy to run compared to any prior D&D. The cost for that, as some people have mentioned, is a loss in granularity, but the gain is that I think I could pick just about any four people, sit them down around a table with pregens, and we could have a fun, functioning game going in a very short time.
The luck part is that the introduction of 5E coincided with an explosion in the popularity of livestream RPG's. There's a little chicken and egg here I'm sure, but it seemed like, as soon as 5E was a thing, everyone was doing it online all the time, which has made it extremely popular.
It help(ed)(s) that Adventurer's League took a lot of good lessons from Paizo's Pathfinder Society and, while it has its issues, it's a great way for people without a regular group to participate. Pre-COVID, there were lots of AL tables where I live. I ran two or three games a month and there was always new people just getting into the hobby. And walking newbies through the basics of 5E was a lot of fun (certainly more fun than the heavy-cart-up-a-steep-hill experience of teaching 3.5 or 4E).