In the actual fact, the moon is constantly [falling] toward the earth.
Because of gravity exerted on the moon by the earth, the moon is being pulled to fall towards the earth. But the moon is in motion and has already acquired a certain velocity (just contrast to the apple example, the apple has no velocity initially relative to the earth), the result of the pullling force by gravity and the initial velocity makes the moon to fall in a curved path, similar to the path traversed by a throwing stone in air.
It so happens that the curvature of the falling path of the moon matches exactly with the curvature of the spherical earth. Because the two curvatures are in parall, the consequence is that the moon could never fall to reach the earth surface, instead it maintains at a constant distance from the earth during its falling.