What is the importance of Polonius' death?
The death of Polonius is central to the play. For one thing, it is the first of many deaths to occur, and it is the cause of many of the other deaths. Claudius is frightened when he learns that his mad stepson has killed Polonius. He tells Gertrude, his informant, "It had been so with us, had we been there." Hamlet was already commissioned to go to England as the King's ambassador, but this murder makes Claudius decide to have him executed when he gets there. This leads to the deaths of Rosencrantz and Guildenstern, who are carrying the letter requesting the execution. But Hamlet substitutes another letter requesting the execution of Rosencrantz and Guildenstern, and then he returns to Denmark, having been kidnapped by pirates and ransomed.
Polonius's murder causes his daughter Ophelia to lose her mind and his son Laertes to start a rebellion. Ophelia eventually dies by what is thought to be suicide, and Laertes is persuaded by the King to kill Hamlet with a poisoned fencing foil. Laertes himself is killed when the two duelists switch foils. Claudius has prepared a poisoned drink because he still wants to make sure of killing Hamlet, but Gertrude drinks it inadvertently. Then Hamlet kills Claudius when Laertes informs him that he is holding an untipped foil with a deadly poison on the point. Hamlet himself dies of a stab wound he has received in the duel with Laertes.
The death of Polonius precipitates the deaths of all the other major characters except for Horatio, who remains alive in order to explain the complex chain of events as best he can.
Polonius's death is a pivotal moment in the play. After all his careful planning and determining that Claudius actually did murder his father, Hamlet, ironically, makes a fateful, impulsive mistake when he stabs Polonius behind the arras, thinking it is Claudius hiding there.
This sets into motion the second part of the play. A very frightened Claudius, knowing full well that Hamlet meant to kill him, not Polonius, sends Hamlet off to England. Claudius puts letters in the hands of Guildenstern and Rosencrantz, Hamlet's companions on the trip, saying Hamlet should be killed. Hamlet turns the tables, has the courtiers killed, and heads back to Denmark, where he finds Ophelia dead. Laertes turns up too, hot-headed and determined to kill Hamlet to avenge his father's death.
Laertes acts a foil to Hamlet. His is a parallel story: he too has had a father murdered and needs to avenge the death. Laertes, however, shows none of Hamlet's introspection or indecision: his is a straightforward desire to get revenge. Claudius is able to manipulate Laertes's hot-blooded anger to get him to agree to the fight with the poison-tipped sword that leads to the last, fatal denouement.
Without Polonius's death, the rest of the tragedy probably would not have played out as it did, as such an utter bloodbath leaving two families (Polonius's and Hamlet's) dead.
This is the tipping point for Claudius. He has to do something about Hamlet who is not becoming erratic and, in the King's view, a threat for the throne. So he orders "The present death of Hamlet. Do it, England" (IV.iii.69)
It also leads directly to Hamlet's (and Gertrude, King, and Laertes) death because Laertes must avenge his father's death. He publicly challenges Hamlet and secretly colludes with the King to poison the drink and the sword.
Finally, it is the breaking point for Ophelia. She cannot live in a world where her love killed her father. Regardless of whether she committed suicide or not, her mental state is too fragile too allow her to survive. "Poor Ophelia/Divided from herself and her fair judgment" (IV.v.86-87).
Simply shakespeare wanted to end the life of his tragic hero dramatically (being killred by laertes avenging the murder of his own father) so to be a perfect shakesperian tragic hero Hamlet must commit a fatal sin that brings about his downfall which is killing polonius accidentally.