Polonius had to be killed in this scene. Otherwise he would have reported to the King that Hamlet was not mad, that he hated Claudius, that he believed Claudius had murdered his father, and that Hamlet was having communications with his dead father. In this scene Hamlet releases all the emotions he has been holding back. He stops being a mama's boy and gains the ascendency over his mother. (In Act 1, Scene 2 he had said, "I shall in all my best obey you, madam.") He makes her promise not to sleep with Claudius anymore and also to keep his secret that he is not mad. Gertrude becomes in effect a co-conspirator, since she must realize that Hamlet intends to kill Claudius and take the throne. When Gertrude goes to see Claudius at the beginning of Act 4, Scene 1, she tells him untruthfully that Hamlet is "Mad as the sea and wind when both contend/Which is the mightier." She holds back much of what transpired in her meeting with her son and says nothing about his seeming to see and talk to the ghost of his father. Although she cannot believe Hamlet was talking to a ghost, Claudius would have believed it because it would explain why Hamlet seems to know so much about his fratricide.
There has been much speculation about why Hamlet hides the body of Polonius. He must realize everybody will be searching for it. He doesn't merely remove it from his mother's chamber but actually hides it where he tells Claudius, "...you shall nose him as you go up the stairs into the lobby" (4.3.35-6)--evidently a considerable distance from the Queen's suite.
A story's climax is a point, not only where the suspense or drama reaches its peak, but also the point that sets of a chain reaction of events that follow (usually referred to as the "falling action").
What makes this scene the play's climax is the dramatic, accidental killing of Polonius, who is hiding behind the curtain. Hamlet is scolding his mother and trying to trigger her guilt for marrying Claudius when he hears Polonius behind the curtain. Assuming it is his uncle Claudius, since he is in his mother's bedroom, he stabs Polonius without thinking.
This is Hamlet's point of no return. He has made a fatal mistake that sets off a snowball effect of events to follow. It is the first act of violence that young Hamlet commits and it is because of this rash act that the rest of the play unravels. Ophelia drowns herself because not only does Hamlet not love her, but he has just murdered her father. Laertes, distraught over the sudden deaths of both his sister and father, challenges Hamlet to a duel and deals Hamlet his deadly blow at the end of Act V.