How researches know about distance change of moon to earth? The method is by beaming laser from earth to reflector in the moon, then they count how long laser takes to reflect measured at a time laser is beaming.

Detecting moon movement has been possible because of photonics technology and Apollo experiment that still running. For 40 years, scientists have periodically pinged laser beam and measuring ping time.

The result, if you want to know, is 2.5 seconds, varying because moon's elliptical orbit results constantly changing distance. From the precise measurements, researchers have confirmed not only key tenet of Einstein's theory of relativity, but also have shown that the force of gravity is very stable and have discovered that the moon probably has a liquid core.

For almost 20 years from 1960's, astronomers measured distance from moon to earth by beaming laser from telescope of McDonald Observatory to retroreflector placed on the moon

Measuring the distance from earth to the moon is dependent upon retroreflector left on the moon surface by Apollo astronauts and by two Russian robot vehicles. Researches on earth beam laser pulse to reflector. The retroreflector sends the laser pulse back to earth. Usually only a few photons make it back and detected to provide the total trip time. That and the speed of light results the distance to the moon with accuracy of a few centimeters depending on the equipment.

Today, measuring moon distance to earth is done by observatories in US, New Mexico and France.

Source: Photonics.com