There isn’t anything strong pushing the moon away, so gravity would pull it down immediately if it wasn’t moving.
It is moving sideways fast enough that gravity just makes it move in a circle instead. If you throw a ball, it goes in a curve until it hits the ground, if you throw it faster, it goes further before the curve that the surface of earth has and the curve the ball follows meet. If it goes fast enough, the curve of the ball never runs into the earth, and comes all the way back around.
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