What happens in III.iv (the closet scene)? Why is this death so important for the play, or what does the death of this figure represent? Act 3, Scene 4
This scene does not take place in a closet, but Hamlet runs Polonius through as Polonius hides behind a curtain, eavesdropping on a conversation between Hamlet and Gertrude. Hamlet, believing that Claudius is behind the tapestry (arras, as it would be called at the time), kills the figure.
Symbolically, this is a representation of Hamlet's dsire to finally ACT as opposed to talk. He is placed under guard and (ostensibly) sent to England, which derails his decision momentarily, but upon his return, the action significantly picks up in the play: his struggle with Laertes during the funeral of Ophelia, and his fencing match with Laertes which leads to the death of all the principal characters in the play.
The most offensive (in his obtuseness), boring, and longwinded character in the play has been killed. Shakespeare is telling the audience that business is about to pick up.