What evidence is there that asteroids collide with planets?

craters and now video evidence.

Most bodies in the solar system with a visible solid surface exhibit craters. On Earth we see very few because geological processes such as weathering and erosion soon destroy the obvious evidence. On bodies with no atmosphere, such as Mercury or the Moon, craters are everywhere. Without going into detail, there is strong evidence of a period of intense cratering in the solar system that ended about 3.9 billion years ago.