So why doesn't the moon rotate like Earth but a bit slower?
You have been told that the moon does rotate, but not like the Earth due to a phenomenon called tidal locking.
So to complete the final part of this jigsaw.
First why does the Earth (or moon or other body rotate at all?)
Well during and after the formation of any substantial body in space other (smaller) bodies crash into it.
Whether they coalesce with the body or just bump into it they must hit it off centre which means that the collision exterts a moment about that central axis.
If all the collisions are from random directions they will tend to cancel out over time.
But if they are mostly from certain directions, as in the early solar system,
over time this builds up to generate a sizable rotational speed and there is not much in space to cause friction so the rotation will last a long time after the period of planet building has occurred.
So we have a body like the Earth, which is slowing down, rotating about its axis, but having substantial rotational momentum
The density of the Earth is
551 5510 compared to the Moon which is only 334 3340 kg/m3. Edit see post7
Coupled with the huge size difference between the two bodies this leads to a huge difference in stored rotational momentum.
I will leave you to calculate the difference.
As strange noted, so far gravity is not involved, but introduce that into the situation and you enter the interaction between two spinning bodies via the tidal locking mechanism when the Moon became captured by the Earth's gravity.
Edited by studiot, 20 March 2017 - 10:40 PM.