Could you explain how a moon draws angular momentum from a planet? I know that the gravitational force transfers momentum, but I don't understand the mechanics behind it.
There's a correct simple answer, a wrong simple answer, and a detailed correct answer.
The wrong simple answer is that the Moon raises two bulges in the oceans. The Earth's rotation pulls the bulge closer to the Moon ahead of the Moon angularly, and this in turn results in a transverse acceleration of the Moon. That transverse acceleration in turn causes the Moon to recede. That's a nice short and simple answer, but it's wrong. The tidal bulges don't exist. These tidal bulges is one of the few things Newton got wrong. What's worse, the people who promulgate that explanation know that it's wrong.
The detailed correct answer is that the tides dissipate in a few key spots on the Earth: The North Atlantic, Patagonia, the coast of Alaska and Australia. Averaged over time, the water piles up in those spots, and it does so in a way that leads the Moon when those spots are closer to the Moon than is the center of the Earth. It's a rather ad hoc explanation and it depends very much on the shapes and alignments of the continents. Right now there are two huge north/south barriers to free flow of the tides, the Americas and Afro-Eurasia. The tidal losses are much higher than nominal because of this, which makes the Earth's rotation rate slow down considerably faster than nominal, and also makes the Moon's recession rate be much faster than nominal. At other times in the Earth's history, the continents were aligned differently and the tides had a much freer flow. The slowing of the Earth's rotation rate was less than nominal during these times, as was the Moon's recession rate. Scientists can see these variations in the changes in length of day in tidal rhythmics and in iron bands. Sometimes length of day changed quickly, other times, not so quickly. The tidal bulge theory doesn't explain this. The dynamic theory of the tides does, but not nearly as simply as the bulge theory.
The correct simple answer is truly simple. Over the millennia, the tides are slowing down the Earth's rotation rate. That angular momentum can't just disappear because angular momentum is a conserved quantity. The angular momentum has to be transferred elsewhere. Since it is the tides that are slowing down the Earth's rotation rate, and since it is the Moon and the Sun that are the causes of the tides, that angular momentum must be transferring to the Moon and Sun. Since the Moon dominates over the Sun by a bit more than a factor of two with regard to tidal forces, most of that angular momentum has to be going to the Moon. The only place it can go is the Moon's orbit; the Moon itself is tidally locked to the Earth.
Does the exact mechanism really matter? One of the big joys of the conservation laws is that they let you skirt around details. By way of analogy, suppose you are trying to read up on Lie groups on the internet. If you follow link after link after link you'll eventually find yourself reading an article on the 1956 New York Knickerbockers. Those details just get in the way of understanding. That angular momentum is conserved and that the Moon is responsible for most of the slowing of the Earth's rotation rate is a good answer.