In the case of proofs by induction, the reason it may be easier to prove a stronger result can be simply that one can use a stronger induction hypothesis.
(I think of the example of proving Łos's theorem in model theory. It says something about formulas that may have free variables. It's proved by induction on the formation of first-order formulas. Imagine trying to prove it only in the case of sentences without free variables. A weaker statement. The proof by induction doesn't work for that case.)