It's something you would have to be keeping up with as a mathematician, really.

Vaguely. It's describing connections between a number of other mathematics results than can be connected to prove FLT. I assume all the work described is being done to make the proof more presentable, smaller, basically "prettier".

It sounds like they established a minimum and maximum bounds for n in x^n + y^n = z^n, where one proof works for n greater than or equal to 17, and another proof for n < 37 (when prime).

I believe the case (remembering back 40 years here) n is even is very easy, and n is composite and odd slightly less so. Neither really being in the ballpark of what they describe here.