Why do fleas live on the furs of cats and dogs?

Fleas require a supply of fresh blood in order to reproduce; the proteins in the blood provide the necessary fuel for them to make their eggs.

Generally they don't live in the animal's fur, they tend to be transient, hopping on board, staying with the animal long enough to establish their food supply, then dropping off and finding a tiny crack to grow and lay their eggs. It can take them a few days to get the amount of blood they need, but most of the time they are actually hiding elsewhere.