Will I need to have access to game materials for the different games I might play? Do I need to keep track of all my own abilities and what they do etc? It would make sense, but I didn't see anything about it in the FAQs.

Like if I want to play a game in 5e will I need all the 5e manuals?

There's nothing about it in the FAQs because it will vary from game to game.  Every GM has their own requirements for their game.  The information regarding those requirements for a specific game can usually be found in the game itself.

BBR pretty much nailed it, but as a general rule, it helps a great deal to have the core rulebook of the game you want to play in either PDF or physical form.  If the game focuses on a particular set of companion rules, you'll likely want those too.

That said, there are hundreds of games you can play (non freeform ones at that) on this site without ever buying anything.  Lots of game systems have what's called a "system resource document" which covers the basic rules that is free.

There's lots of games I've started without having anything, but if I like the game I usually buy a copy of it eventually to support the creators.

I've never played a game on this site where I needed absolutely all of the books for any given system.  I think the most books I've ever "needed" was three, and they were games I collect anyway and have tons of books for.

I don't think the rules let me mention the exact site, but you'll find it easily with searches.  There is a certain popular and reputable site that often has sales on pdf copies of various games.