No, there is no intuitive way to communicate it. The theorem has taken >350 years to prove, which makes it clear that the proof is not some intuition that was somehow missed by hundreds of people for centuries.

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.

Thanks for the recommendation good sir!

Well, the answer is "simple": It's because the modularity theorem was proven (or better, the Tanyiama-Shimura conjecture was proven)

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 still like the idea that Fermat had a legit proof and that one day a simpler one will be found.

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.

If there were a simple and intuitive way to communicate the answer then I would suggest that we probably would have figured it out in our 300+ years in which this was one of the most famous unanswered questions in mathematics.