Of course the consequences of that theorem are legion, and they do warrant many books worth of discussion. But the proof itself can fit in a long-ish blog post, I think.
Fermat's last theorem, on the other hand...
Assume a formal language that can model the natural numbers.
Consider the set of all provable statements. This is countable because it’s a subset of the list of strings of a finite number of characters, which is countable.
Consider the statement “Statement N is false”. This is both a valid statement and cannot appear in the list.
Godel proved significantly more than that, but that’s the basic result. It’s basically the same as the diagonal argument of real numbers being uncountable and it’s also basically the halting problem.
The crucial trick is that you can plug in an entry as data to another "program/theorem". This is extremely interesting, because there are limited logics and limited computation models where Godel and the Halting Problem do not apply.
Not so sure about the diagonal argument.