When students ask me for the use of the formal and abstract theory, I often would like to give answers they wouldn't understand. For instance, one application of abstract vector spaces and the banach fixed-point theorem is solving differential equations by applying the theory to function spaces. That's too much for a beginner.
A nice one is what is called the "football theorem" in german (Satz vom Fußball). Compare a football lying on the center spot before the game starts to the same football lying on the center spot before the 2nd half starts. There will be (at least) 2 point which are exactly in the same position, how ever much the ball was turned during 1st half. The proof is given by the fact that the composition of rotations still is a rotation having an eigenvector in three-dimensional space.
Is there more like this?
Edit: I think of first-year undergraduate students which are confronted with formal definition and proof for their first time and ask "why?".