On average, how much further away from the Earth does the Moon move each year?

The answer is actually about 3.8 centimeters according to most lunar measurements.

The reason for this is centripetal force, offset by earth's gravity.

How do they measure this phenomena?

Since the early 1970's, scientists have been bouncing lasers off the moon (off plates that were left by the Apollo missions) and measuring distances and changes in those distances on a steady basis.

It was only in the early 1990's that scientists had been able to put together a theory about this by measuring the thickness of tidal deposits preserved in rocks, called tidal rhythmites, which can be billions of years old, although measurements only exist for rhythmites that are 900 million years old.

The bottom line is that in the next 5 billion years, the moon will appear about half the size it appears today.