What keeps the earth spinning?

The solar system of which the planet Earth is a part, originally formed out of a large cloud of interstellar gas. The cloud had its own spin, and that angular momentum was given to the various objects which condensed out of it.

If you wish to trace this back even further, and ask why the original cloud was spinning, we can go back to the Big Bang, an explosion which formed the universe as we know it. The original momentum of the Big Bang was entirely linear; everything moved away from the center of the explosion. However, linear motion is easily warped into angular motion, by gravitational interaction.