While Earth does contain craters on its surface, they are not nearly as visible as those on the Moon for a number of reasons. Earth has a molten center and plates that move as a result, and the Moon does not; Earth is covered in water, and scientists are still trying to conclusively find quantities of water on Moon’s surface; and Earth has a weather system that does not occur on the Moon, where the former’s surface is quickly eroded by rain and wind, while the latter’s surface is more slowly eroded by meteoric bombardment.
The first reason listed, plate tectonics, occurs on the Earth with one of the byproducts being Earthly craters are “covered up” by moving plates. With no molten core in the Moon, there are no moving plates and its craters are left exposed.
Secondly, Earth contains a great deal of water on its surface, causing the hydrologic cycle. Rain falls with enough force on the craters and the areas surrounding the craters that, over time, the craters become eroded to a smoother and more level plane. With wind also being part of Earth’s weather system, the air strikes the craters and surrounding surfaces hard enough that it causes erosion. On the Moon, the erosion that occurs is by meteoric bombardment, a much slower process when compared against rain and wind. According to Eric Chaisson,
Most craters formed eons ago as the result of meteoritic impact. …impact by a meteoroid causes sudden and tremendous pressure to build up on the lunar surface, heating the normally brittle rock and deforming the ground. The ensuing explosion pushes previously flat layers of rock up and out, forming a crater. (p. 153).
Chaisson, E. & McMillan, S. (2011). Astronomy: A Beginner’s Guide to the Universe.
(6th ed.). Benjamin-Cummings Publishing Co.