Why do people believe in evolution?
People believe evolution because it is a theory based on science and observed in nature. The mechanisms of evolution are still a point of contention, and it remains a theory (or one possible solution) to species diversification.
The idea of evolution has been around for centuries and was popularized in 1859 by Charles Darwin and the Origin of Species. He postulated species would adapt to niche environmental opportunities. His research led him to believe all species share a common ancestor. It is a widespread myth that he believed man evolved from apes.
Phylogeny is the study of inherited species' characteristics based on historical evidence. In essence, it is reading the family tree of a species to understand shared ancestors. A clade is a grouping that includes all of the shared species to a single ancestor. For example, species (A) exists and at some point in time there is some mechanism encouraging change in the species. Species (B) develops from (A), perhaps because a different type of wings. The process repeats with species (B) and divides into species (C). Now there are three species (A), (B) and (C) which are all related.
People believe in the theory of evolution for the same reasons people believe in the theory of creationism. There are scientific facts and observable indices in nature that support the theory.
I would also like to say that when this question is framed as a matter of "belief," it is being framed in a disingenuous way that sets up a false equivalence between science and religion, an equivalence that is prejudicial and completely without merit. Religion is a matter of belief. Belief means that one subscribes to an idea with no evidence whatsoever. I can believe in a deity, not because there is tangible evidence of one, and there clearly in none, but because I choose to so believe. But we do not believe in evolution. We are persuaded by the evidence, as the first response indicates. Scientists may have beliefs, too, but they do not present them as scientific facts or theories; they understand that they must explore, investigate, research, and experiment. They must find evidence. No scientist would ever ask if someone believes in evolution, and no belief requires any research. Sometimes how we ask a question is important, because it can set up the answer we want to have.