How do we prove that parallel light rays (which are not parallel of principal axis) meet at focal plane?

The proof is “by construction”: the lens, mirror or more complicated optics system was constructed to do exactly that.

Why is that useful? It maps light arriving at each particular angle to a specific point on the focal plane. Since light from different points on a distant object arrive at different angles, this lets the optical system form an image of the distant object.

you put a screen behind the lens, and you will see the light make a small spot as you move the screen. There lies your focal point.