"핑계 없는 무덤 없다" == 핑계 없는 무덤은 없다. => There's no grave that doesn't have a reason. => Every grave has a story. 핑계 normally means "an excuse", but here it's used more like a reason or cause that brought about something. That is, there is no grave(a dead person) that did not have a (long and involved) story behind it. It is a proverb saying everything has a cause that led to it, or that nothing in this world happens by itself in isolation. I also like to think of it as "Each person is like a world unto himself with so much story behind him", but that might be taking it too far :-)

It basically means "Everyone's got his/her excuse" as every grave has a story(a story -presumably- claiming its owner's untimely or unjust death).

It means.. Always there are something reason whatever happen.