A big part of it is also pattern recognition. Recognizing what the various markers mean, and how to react to them. Like, a stack marker means the same thing in every fight you see it; you don't have to relearn it for every fight.

Not to mention, several fights reuse mechanics. If you see a boss turn itself into a ring, or make a ring shape, it's going to do a ring aoe around the arena. If you see a boss raise it's right fist, it's probably going to hit the right side. If you're standing behind a boss and it raises its tail, get out from behind it, because it's probably gonna slam its tail down on you.

I find that these visual telegraphs seem to be the ones who get the most new players, so my other recommendation is to actually keep an eye on the boss, and not just your hotbars. Especially in later expansions, you won't be able to react fast enough to the orange markers to avoid big attacks unless you position yourself in the right place based on what the boss is doing at that moment.