- Newton thought that a force must act on the Moon because, since it moves in a curved (almost, but not quite) circular path, it is accelerating.
- The gravitational force is an inverse-square-law force - the strength of the force decreases with the square of the distance between objects.
- One of the laws of motion that Newton figured out was that an object in motion will tend to stay in motion, IN A STRAIGHT LINE, unless acted on by an outside force. (A lot of people forget the straight line part.)
So, since Newton knew that the moon was moving around the Earth in more or less a circular manner, and not just passing by in a straight line, he knew that there had to be a force acting on it.