The real basic nuts-n-bolts explanation for Fischer appears (to me) to be just that he's a nut case. Even among international chess competitors where that's somewhat "normal" he was (is?) a bit bizarre.

Wikipedia has a pretty full history.

I haven't been able to tell from news reports just how much of an "in-charge-guy" Kasparov is in the recent protests, but his name name and reputation should give him some "voice" - one would hope. If he's not the leader he's certainly likely to be high among the leaders. The protest efforts seem to be centered among members of the legal professions(?) but details have been sparse.

I just hope he'll keep his head down and stay safe - and wish him well.

John