How does the sequence in the fossil record provide evidence for Darwin’s theory?

There are a variety of ways to interpret the fossil record in a way that supports Darwin's theory.

One of the primary considerations is the Law of Superposition, which says that, in the absence of disruptive forces such as plate tectonics altering the orientation of fossil layers, the oldest material should be on the bottom, and the newest material on the top. This is typically demonstrated upon observation through several means, such as radiometric dating, as well as observing the nature of the fossils themselves. If the Law of Superposition were not in place, we would expect to find human skeletons in one-billion-year-old rock, but instead we find bacteria and nothing else. This supports the idea of evolution in that it demonstrates how life "builds upon" earlier forms. Note that we should be careful to avoid the terms "improving" or "getting better" because these are relative and imply qualitative judgement based on our own anthropocentric stereotypes; bacteria are just as good at "being alive" as humans are, they simply lack the structural complexity that we have, and that complexity should, logically, develop from simpler structures, rather than the other way around. The fossil record directly demonstrates that "simple" species, such as bacteria, do not descend from "complex" species like humans, and that this trend appears to be consistent for the entire history of the earth. 

Another way the fossil record supported Darwin was the existence of extinct species. This demonstrates that not all life is permanent and that competition can definitively lead to the complete death of life that is not suited for its environment. What was most definitive in Darwin's time was the fact that fossil species which had never been observed as living animals were being discovered, demonstrating that species had been experiencing these forces long before human-recorded history.