Earlier quoted context omitted.
One of my advisors in college said "The proof takes about 10 years of graduate mathematics to understand".
Is there really no intuitive way to communicate the answer to that question without needing 10 years of grad math? I find that to be somewhat hard to believe
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.