Why does earth's gravity pull the moon tworads it?

It is the nature of the gravitational force to attract things. The moon is in free fall around the Earth, as is the Earth around the Moon. They don't collide because they have significant angular momentum. This causes them to 'miss' each other constantly.

However the Moon is actually receding from Earth. This is because the tidal motion on Earth robs the Moon of energy so its orbit is increasing. At the same time the speed of Earth's rotation is also decreasing. In the end this effect will stabilize with the Moon's period of rotation around the Earth being equal to a day on Earth.