Why do fossils prove that evolution has occurred?

Fossils themselves prove nothing.

However, in comparing fossils from earlier, later and much later times, we see such similarlities accompanied by differences; we see creatures remaining as they were, and then suddenly developing into others, sometimes disappearing, soemtimes living alongside them.

The fossils show us a small picture of basically what was alive at that time. We know that for example, some apelike creatures existed in one place at one time, and then later on they moved "here". Then we see that they start becoming fewer and fewer, while a similar - but different - creature becomes more common. Then #2 becomes rarer while another very similar but different animal becomes more common. And then #3 starts getting less common...

Sometimes at each stage, we find there are several animals which are all different, but all similar to the ones before. Sometimes we only one, immediately replacing its predecessor. As explained by evolutionary theory, this is what we expect, and this is what we see.