Hamlet, who speaks the majority of the lines in the play, is about to die and stop speaking for ever. He has avenged his father's murder, been cut with the poisoned rapier, and, as he has just reflected to Horatio, is about to pass the crown of Denmark (he is, at this moment, effectively the only candidate to be king) to Fortinbras. There will be no familial succession. In that sense, the rest is silence for the line of the Hamlets.
Yet there is (as with so many other things in Hamlet) a meta-theatrical element. Hamlet has just asked Horatio not to commit suicide (Horatio's line 'Here's yet some liquor left' refers to a drop of poison left and his desire to kill himself) but to stay alive and to 'report' Hamlet's story to Fortinbras.
Hamlet cannot voice his choice for succession nor can he tell his own story: he is about to die; there can only be silence from him. But when the play--itself a version of Hamlet's story--comes to an end, part of Horatio's project, even after the curtain has come down, is to re-tell that same story:
So tell him, with the occurrents, more and less,
Which have solicited.
Paraphrase: So tell Fortinbras, with the events and results that came to pass, more or less, which wrongdoing was provoked and incited.
So, perhaps, as we come to the end of a performance of Hamlet in the theatre, we confront Hamlet. Yet, in an odd sort of way, we simply come to the end of a cycle. There is silence at the end of each performance, but every time the play itself is performed, Hamlet's dying wish for his story to be told 'aright' is fulfilled.
Horatio, I am dead;
Thou livest. Report me and my cause aright
To the unsatisfied.
O good Horatio, ...
If thou didst ever hold me in thy heart
Absent thee from felicity awhile,
And in this harsh world draw thy breath in pain,
To tell my story.