[dreamcompiler]

Great question. The answer is nothing. There is no force making the moon follow a circular path. The moon "thinks" it's moving in a Newtonian, inertial "straight line." Because the spacetime around the earth and moon is curved, the moon moves in a straight line through that curved space.

Caution: The circular path we see the moon follow is not the curvature of spacetime itself. Rather it's a zero-force iso-line along that 4D spacetime. This is also called a geodesic.