What is the importance of the gravedigger scene in the story of Hamlet?

It's an absolutely key scene. Shakespeare is an absolute master at juxtaposing the comic with the dramatic: and the two gravediggers' comic dialogue which precedes Hamlet's entrance prepares the ground for real surprise when the same grave, only a few hundred lines later, becomes the site of an impassioned fight between Hamlet and Laertes.

It's often the name of the game in Shakespeare's "comic relief" scenes that the themes reflect in a comic way the wider themes of the play. Thus, the opening argument about whether Ophelia "drown'd herself wittingly" reflects the play's concern with suicide - as outlined earlier in the "to be or not to be" soliloquy.

The scene also achieves precisely the absurdity which baffles Hamlet: the juxtaposition of death with comedy. It's no accident that the gravediggers are designated as "First Clown" and "Second Clown". Thus we have jokes about the pre-eminence of death: the gravemaker's houses "last till doomsday." And death even visits Yorick in the scene - an absent clown, now just a skull, who Hamlet and the gravedigger remember as "a fellow of infinite jest, of most excellent fancy". Even funny men - even men who tell jokes - die.

It's a brilliant scene which focusses the theme of "death" ready for the play's final scene and Hamlet's death - and provides the final comic moment of the play before the denouement of the tragedy.