For much of the play Hamlet is terrified of dying, or maybe more accurately terrified of what may happen after death. “No traveler returns” from death, he muses, forgetting in that moment that his father HAS returned, at least possibly, assuming, of course, he’s not a demon in disguise… What happens to your soul, your consciousness, after death consumes him – what if it’s torture? What if it’s nothing? The possibilities trap him and leave him unsure how to proceed.

Something has happened to him by the time he encounters the Gravedigger, though. Having been unable to kill Claudius, he was shipped off to England, ensured the death of two colleagues, and came extremely close to death himself at the hands of pirates. He escaped, barely, and the brush with death has changed him. The skulls don’t repel him (well, they still do a little bit), instead they remind him that everyone will be a skull eventually, and the thought of that leaves him oddly peaceful. Rather than a terrifying plunge into the unknown, he starts to see death as the last stop on a journey, to be embraced and acknowledged as inevitable.

Hamlet has spent the play fighting against death and struggling with his own feelings about it.  However, in this graveyard seen, he seems finally to have found some peace.  He acknowledges the inevitability of death, and accepts that it is the natural end to everyone's life, rich or poor, important or insignificant; it binds all humans.  Unfortunately, at the moment at which he might have given up his vendetta, Hamlet is propelled into action, which then leads to his own death and the death of many around him. 

When Hamlet first encounters the gravediggers, his reaction is one of disbelief at the treatment of the dead:

Has this fellow no feeling of his business, that he
sings at grave-making?

He goes on to say the following:

That skull had a tongue in it, and could sing once:
how the knave jowls it to the ground, as if it were
Cain's jaw-bone, that did the first murder! It
might be the pate of a politician, which this ass
now o'er-reaches; one that would circumvent God,
might it not?

Hamlet is questioning the actions of the gravedigger, but here, one can see that he is starting to think of humanity and the fact that all of these skulls are human and were living once. He's beginning to see that the skulls represent what everyone becomes once they die.

What really makes him realize this is when he sees Yorick's skull. Because he knew Yorick, he can make the connection to the rest of humanity and himself. Eventually, he references Alexander the Great asking:

Dost thou think Alexander looked o' this fashion i'
the earth?

This suggests a change in his outlook in that he's realizing that death comes no matter what. He's starting to see that his life isn't worth preserving and that he can and must risk it in order to avenge his father's death. Once he sees that the gravediggers are digging Ophelia's grave, he has nothing to live for.