What makes the Earth keep spinning?
I agree with the previous answer. Angular momentum, something the earth has because of its rotation about its axis, can only be changed when an external torque (twisting motion) is applied to the earth. As far as I know, there are two ways in which this can happen. If there was friction between the earth's surface and space, then that would slow down the earth. However, this is negligible because space is essentially a vacuum. The second way is through the gravitational/tidal force applied by the moon. This force creates the tidal bulge of the earth's oceans, exerting a torque. So, while the earth seems to be spinning at a constant rate, it is actually slowing down slightly over time because of these tidal forces.
This isn't really an answer, but it's related and it got a bit long to put in as a comment.
Steven Harris has mentioned that the Moon is gradually slowing the Earth. The obvious evidence for this type of change is that the earth has slowed the Moon's rotation until it matches the Moon's orbital frequency - so the moon always keeps the same face towards the Earth. However the earth produces bigger tidal forces on the Moon that the Moon does on Earth, so the Moon is affecting the earth's rotation far more slowly.
According to Wikipedia the Earth's rotation is slowing at about 1.7ms per century. A quick bit of arithmetic shows that if this rate stayed constant (and it probably slows over time) it would take about 5 billion years to bring the Earth to a standstill. Hence the comment that well before this happened the Earth would have been fried by the Sun's red giant phase.