In this context it means the earth is falling on the coffin making tapping sounds. A good example is "the rain pattered on the window" - it is making tapping noises on the window. :)

"Pattering" is a light tapping sound. People will use the word "pattering" when they describe the sound that a light rain makes or the sound that children make when they walk. The author is describing a scene at the burial. Some people have a tradition where they have the funeral right next to the place the person will be buried. When that happens, they dig the hole before the service (the day or week before), and leave all of the dirt right next to the hole. At the actual service, the mourners show up. They bring the coffin (dead person inside) to the hole, they put the coffin in the ground, and then they fill the hole with the dirt that they'd previously dug out. Some people have a tradition where they do not leave the funeral until all of the dirt is back in the hole. (Some people have a tradition where family members and close friends pick up a handful of dirt and put it on top of the coffin (which is in the hole). Some people have traditions where they wait for the person to be partially buried, but leave before it is finished. Some people don't have a tradition at all.) The author is saying that when his mother died, he was at the grave while they started putting the dirt back in the hole. When they did that, it made a sound and he would describe the sound as "pattering." When the author looked at the pile of dirt, it was red dirt but there were pieces of tree root that looked white. (If you've never seen roots mixed in dirt, I can't help you there)