Earth is more active, in terms of both geology and weather, which makes it hard for craters to remain. Even those craters scientists can see on the surface—which may be millions of years old—have been overgrown by vegetation, weathered by wind and rain, and changed by earthquakes and landslides. The Moon, meanwhile, is geologically quiet and has almost no weather, so its hundreds of thousands of craters are easy to see.
The craters are the result of both meteorites and volcanic activity. Interestingly, some of the oldest Earth rocks might be awaiting discovery on the Moon, having been blasted there billions of years ago by asteroid impacts that shook both worlds.