When you tie a rope to a rock, and swing the rock in a circle around your head, the same thing happens.
That is, the inward force (you pulling the rope, gravitational, electrostatic, etc) is necessary to make the object travel in a circle. Anytime you see circular motion, there must be an inward force that always points tward the center of the circle.
If the moon was made to have zero speed at this instatn, it would begin to fall to earth, gradually at first, then gaining speed (just like you tugging an initially stationary rock). It is the net inward force + velocity around the circumfrence (perpendicular to the inward force) that are necessary and sufficient for circular motion.