This means understanding what Modular Forms and Elliptic Curves are in their own right. Then trying to understand Wiles’ proof that they are the same thing.

The keys were the Taniyama/Shimura Conjecture, which guessed that Modular Forms and Elliptic Curves where the same thing. Then in mid-80s, Frey and Ribet (from Berkeley – Go Bears!) proved that if Taniyama/Shimura was correct, Fermat’s Last Theorem would also be true.

http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html