Why does rainwater form moving waves on the ground? Is there a name for this effect?

These waves are called "roll waves." They are due to an instability in shallow shear flows. The analysis is much too complex for a short answer, but if you google "Roll Wave" you will find more images and links to technical articles. If you are not bothered by a little mathematics you will find a discussion of the cause of the instability starting on page 259 in these online lecture notes: https://courses.physics.illinois.edu/phys508/fa2016/amaster.pdf

After the waves have formed due to the instability, the actual form -- a series of breaking waves -- is due to the non-linear propagation effect described by md2perpe -- the deeper the water the faster the wave.

I think that the explanation is that waves of different size run with different speeds. This makes the faster waves run up to the slower which make them stack up, or constructively interfere.