I've found it's basically as complicated as you want to make it. I've never played WoW but I don't have problems getting along in the game; essentially all you need to know is "there's demons attacking and everyone distrusts/hates you and everyone else because of it" and you can run from there. Plus because Warforged is explicitly an AU to Warcraft/WoW canon, even though there's WoW OCs who're somewhat familiar with their world's version of events, they don't really have any noteworthy advantage over anyone else.

And while some of the game's setting info is a bit infodump-y a lot of it hasn't actually come into play yet and is physically distant enough that it doesn't currently affect things short of medium to large-ish plots. It's, well, kind of like an MMO in that regard--when you're new you start out interacting with things in the newbie area and then as you go you can branch out from there as you get the hang of things/your character is so inclined. There's a bigger world out there but you don't have to take it in all at once or anything.