They use laser ranging to determine the distance of the moon (http://physics.ucsd.edu/~tmurphy/apollo/apollo.html). They bounce Earth-based lasers off of reflectors left on the moon by the Apollo astronauts.
"Measurements show that the Moon is receding from Earth at a rate of about 3.8 centimeters per year." (http://eclipse.gsfc.nasa.gov/SEhelp/ApolloLaser.html)
Here is the original paper publishing the results: http://physics.ucsd.edu/~tmurphy/apollo/doc/Dickey.pdf.
This level of accuracy is possible because they are bouncing a laserbeam off of mirrors on the moon. They know how fast light travels. They know the geometry of the beam's path. So, they can work out the Earth-moon distance.