@GP: I'd recommend to try to read an understand the proof for N=3. (And why that approach does not extend to bigger N.) It requires only undergraduate level math and it is much much much easier. It uses very different tools, so it will give you very little insight of the general proof, but it will give you some taste of the problems of the proof.
Fermat's Last Theorem (book) by Simon Singh is the source to check out if you're interested in the details of how it eluded mathematicians and a general idea of how the problem was solved, without getting too technical. It's a great story well told.
But why that solves the problem? Because it connects two branches of mathematics (modular forms and elliptic equations) in a way that proves that equations of that form cannot exist (where the exponent is > 2)
Though there probably is an easier way of explaining it, it is strongly suspected that Fermat got the wrong idea there.
I also like that FLT follows easily from the Beal conjecture, which seems overlooked. Maybe its overlooked because its closely related to some other (harder to understand) conjectures.