In quantum mechanics, the "ball" (or in this case an ideal particle) has a "wave function" associated with it. This wave function effectively describes the probability that the particle can be at a certain location.

It so happens that when you solve for this problem, a ball bouncing against a wall, in this wave function paradigm then you end up with a non-zero probability that the ball appears on the other side of the wall.

I'm not sure if there is a deeper explanation at play here but that's how I understand it.