Hamlet is dying. He will not be alive to make any more noise. That is the simplistic answer. It is more intricate than this. He is too young to die. He probably should have lived a few more decades at least. Those years will not hear his voice or his life. In death, he has been silenced.
Hamlet pleads with Horatio to inform Fortrinbras that he, Fortinbras, is to take the crown. He also asks Horatio to tell his story. Hamlet will not be around to tell anyone of how his uncle murdered his father. He will not be around to explain his anger with his mother. He will not be around to do any of these things because he will be dead. He longs to set the record straight but he will be silent.
He has said all that he can within the limited time he's had on Earth. And we know what a talker Hamlet can be. It is therefore more painful for him to be silenced in this way. He wants to go on talking, to stand up for the reputation of his father and his own. "The rest is silence" indicates that Hamlet will speak no more. He must rely on Horatio to share all of the information he will not live to tell:
But let it be. Horatio, I am dead;
Thou livest; report me and my
cause aright to the unsatisfied. (V.ii.346-48)