Can't say I've heard it said too often, but it seems to me as though this means the same thing as "You cannot hide the past." In other words, the past will always find a way to be known/to reveal itself...it will always "claw its way out" in one way or another.

Animals that can dig into the ground claw their way out from under the soil. The image can be used as an idiom. Hence, if you try to bury the past, it'll claw its way out again.

It's called karma.

The full quote is "You can't bury the past, because the past claws its way out." The image is one of something digging itself out of its own grave. Marg and Kat have good explanations. Another way of say this is "The past has a way of coming back to haunt you." or "His past came back to bite him."

"Because the past claws its way out." Oh, I like the image! Claws are sharp curved nails that predators and beasts like hawks and tigers have. It's difficult to get through life without some horrible things happening along the way. Some of us would rather forget the past. Unfortunately, the past can sometimes suddenly come back and affect the present, or even violently "claws its way [back] out."