Why is the Earth's moon so big?

Most moons are captured bits of rock left over from the formation of the planet as it swept its bit of orbit clear.

Our moon is (probably) the result of a very big bit of rock hitting the early Earth and splitting it into two. The smaller bit ended up as the moon.

This also explains the relatively high density of the Earth (it got left with all the heavy core parts) and the low density of the moon (it got all the light surface rocks)