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.