When I listen to real ATC on liveatc.net, it sounds just as bad. The
quality problems aren't the same, but their magnitude is. I'm
surprised more aircraft aren't running into each other. I can only
hope that the audio quality is much better aboard the actual aircraft,
but I doubt it.

My experience is that controllers are always pretty clear, although
they may talk rapid fire. I think this is because controllers have good
headsets.. not much background noise, and powerful (?) ground based
transmitters. As opposed to pilots who have lots of noise around them.

For someone like me that is more visual than auditory (it sounds like
you are similar) it takes practice to be able to visualize what I just
heard. It helps that for almost everything it follows a very standard
format (and controllers are good at following that format) as opposed
to us pilots that tend to try it on the cool, i.e. "uh, up from 4 for
5.2" etc. Those Don Brown columns I pointed you to earlier has a whole
bunch of columns on phraseology and its usefuleness for safety.

Do you listen to Vatsim on a headset? Might make it easier.