At any one time, there is one side of the earth facing the moon, and one side facing away from the moon. The side closest to the moon feels the moon’s gravitational field more strongly, and is pulled slightly towards the moon. The earth on this side is raised a full centimetres towards the moon. A more noticeable effect is that the water is also raised slightly towards the moon, creating a so-called “tidal bulge”.
Now, this “tidal bulge”, of both the solid earth and the water also increases the gravitational attraction towards the moon.
The earth rotates on its axis much faster than the moon orbit the earth (24 hours for an earth rotation vs 27 days for the moon to orbit). This means that the tidal bulge quickly pulls ahead of the moon- just due to the earth’s rotation.
The gravitational pull of the tidal bulge then tries to speed the moon up in its orbit of the earth, because it is pulling ahead, whilst the gravitational pull of the moon tries to slow down the rotation of the earth. (This is done in such a way as to conserve the angular momentum of the whole system.)
Now, a speeding up moon is gaining energy, and an increase in the moon’s kinetic energy corresponds to a higher orbit around the earth.
(This in turn makes the moon feel the earth’s gravity more weakly, so the moon in turn slows down.)
The moon is leaving us at between 1-4cm per year.
Here’s a really good analogy I’ve thought of. Imagine two ice-skaters, holding hands and spinning round each other, getting faster and faster. As they spin, the centripetal force between them gets faster and faster, and pulls them further apart. Eventually, it will get so powerful, they will be forced to let go and fly apart from each other.
You’re right — the Moon’s orbit is indeed getting larger, at a rate of about 3.8 centimetres per year (out of an orbital radius of around 384,000 km).
The reason for the increase is that the Moon raises tides on the Earth. Because the side of the Earth that faces the Moon is closer, it feels a larger gravitational force than the centre of the Earth. Similarly, the part of the Earth facing away from the Moon feels a weaker gravitational force than the centre of the Earth — you can actually calculate the exact force by making some assumptions about the mass-density of the earth, and integrating over distance. The overall effect is an asymmetric stretching force. Funnily enough, this effect stretches the Earth a bit, making it look a bit like an american football or rugby ball. The actual solid body of the Earth is distorted a few centimetres too, but the most noticable effect is the tides raised on the ocean.
Now, all mass exerts a gravitational force, and the tidal bulges on the Earth exert a gravitational pull on the Moon. Because the Earth rotates faster (once every 24 hours) than the Moon orbits (once every 27.3 days) the bulge tries to “speed up” the Moon, and pull it ahead in its orbit. The Moon is also pulling back on the tidal bulge of the Earth, slowing the Earth’s rotation. Tidal friction, caused by the movement of the tidal bulge around the Earth, takes energy out of the Earth and puts it into the Moon’s orbit, making the Moon’s orbit bigger (but, a bit pardoxically, the Moon actually moves slower!).
The Earth’s rotation is slowing down because of this. One hundred years from now, the day will be a whole 2 milliseconds longer than it is now. This is, I believe, part of the reason there’s a difference in times between GMT and UTC. It’s also vitally important for GPS satellites to be aware of, and is generally a bit of a pain.
This same process took place billions of years ago — but the Moon was slowed down by the tides raised on it by the Earth. That’s why the Moon always keeps the same face pointed toward the Earth. Because the Earth is so much larger than the Moon, this process, called tidal locking, took place very quickly (geologically speaking), in a few tens of millions of years.