|Greater blind mole rat|
There are two broad types of explanation for what drives evolution: they're called the Red Queen and the Court Jester. It's important to note that there's no reason why they can't both be responsible. Indeed, it would be astonishing if they weren't. The debate, then, hinges on which is more important, and when.
The Red Queen hypothesis, first formalised in the 1970s, is named for the character in Through the Looking Glass, who had to run as fast as she could, just to stay still. The idea is that no animal (or other living organism) is alone, and, if you don't evolve, the creatures around you will, and you'll be doomed. One species may be very good at eating a particular type of plant, for instance, but if another species gets slightly better at doing the same thing, if the first species doesn't improve, it will be out-competed and go extinct. You always have to be better than your competitors, and they have to try to be better than you, leading to a constant drive for change - towards better adaptation to your environment and chosen food source, whatever that may be.