xkcd

So the way I "understood" the spin-statistics theorem is that it's basically this:

A given particle with a given intrinsic spin has a direct relationship to a collection of the same particles as a consequence of quantum math. Yeah. Just "it's related."

Proving that math is really freaking difficult and you need to use relativistic quantum field theory. I think it was Richard Feynman who said "We apologize for the fact that we cannot give you an elementary explanation."

Actually when I graduated there was another professor (can't remember his name) who was discussing his frustration with how they still can't explain it without all of QFT steps.

Basically, this is where the shared attitude of "the more you know about quantum physics, the more confusing it becomes."