When Hamlet and Horatio arrive upon the scene of gravediggers or "clowns" digging in the graveyard, Hamlet opens his appearance in the scene saying "Has this fellow no feeling of his business, that he/ sings at grave-making?" He is appalled at the idea that the sexton is so unfeeling that he would be singing. In response, Horatio says "Custom hath made it in him a property of easiness," meaning that he is so used to what he does for a living that he doesn't feel anything when doing it. It's his job and he can't get caught up in being "sad" about it.
The important part of this scene comes later, after Hamlet discovers "Yorick's skull." Yorick was the King, his father's, jester and Hamlet knew him. When he sees the skull of a person he knows, he realizes that all life comes to this in the end. The sexton knows it, which is partly why he is able to sing at digging the graves. Hamlet's realization of this fact allows him to think of his life as less important and in the end, helps him to finalize revenge. He explains it using language about great "ancient" leaders:
but to follow him thither with
modesty enough, and likelihood to lead it: as
thus: Alexander died, Alexander was buried,
Alexander returneth into dust; the dust is earth; of
earth we make loam; and why of that loam, whereto he
was converted, might they not stop a beer-barrel?
Imperious Caesar, dead and turn'd to clay,
Might stop a hole to keep the wind away:
O, that that earth, which kept the world in awe,
Should patch a wall to expel the winter flaw!
Alexander the Great and Julius Caesar now also look like this in their tombs. Everything goes back to the earth. The sexton knows it and now Hamlet is fully aware of it.