This may be a little trivial, but there are a number of identities for the Fibonacci numbers that are most easily proved by generalizing them. For example, proving
$f_{2n-1}=f_{n+1}f_{n}-f_{n-1}f_{n-2}$
requires a rather convoluted process involving a couple lemmas unless one realizes that it is far easier to prove
$f_{m+n}=f_{m}f_{n+2}-f_{m-2}f_{n}$
and then substitude $m=n,n=n-1$.
I think a classic example of generalizing in order to prove a simple result is Galois Theory. Ruffini's attempted proofs of the unsolvability of the quintic were enormously long and tremendously complicated. However, once the machinery of Galois Theory is developed, which is rather easy, it is almost trivial to demonstrate that there exist quintic equations that are not solvable by radicals.
