What happens in Act 3, Scene 4 (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
The death of Polonius, while not the climax of the play is a climatic moment for the audience. Finally, we see Hamlet able to take action! We just came off the disappointing scene of Claudius praying, and while we understand Hamlet's decision to not kill Claudius while at prayer, we are frustrated as well. Hamlet's rather implusive stab through the curtain of a character we haven't liked from the start is rather satisfying; especially in light of Hamlet's lack of remorse. This murder, though accidental, doesn't change the playing field though. Hamlet has now done something dangerous and "crazy" and now Claudius has to do something about Hamlet while not upsetting Gertrude and the whole of Denmark. He can't outright kill Hamlet -- Gertrude would never forgive him. The people of Denmark would never understand. He also knows that Hamlet's knows the truth and is ready to take action, therefore he is fearing for his own life for the first time. The rest of the action of play is, in part, connected to Hamlet's actions in this scene.
There are a couple of ways to interpret the death of Polonius, but I tend to think of it as the death of the prating, foolish idiot and the awakening of Hamlet to the real consequences of his plot to get revenge on Claudius. He too now has blood on his hands, and though it was accidental and he mistook Polonius for Claudius, it still spurs him to further action. His willingness to joke about the man's death and his hiding the body and joking about where he hid it also suggest the incredibly low esteem in which he held the man.
As far as why it is so important to the action of the play, I think the conversation between Hamlet and his mother is far more important as he brings her to the edge of the abyss by reminding her of the love she had for Hamlet Sr. and deriding the relationship she now has with this murderous villain currently occupying the throne. It brings out again the idea that she is duplicitous in the rotten state of Denmark.