In Hamlet, how does the play The Mousetrap in Act III, Scene ii, reflect issues that appear elsewhere?

The play within a play imitates almost precisely what Hamlet suspects has happened to his father and mother, but who told him what happened? His father's ghost! So The Mousetrap is used to verify, or prove to Hamlet, beyond a shadow of a doubt, that his father's ghost was real, for one thing.

Up till he stages this play, he is not absolutely certain the ghost is real, is telling the truth, is anything more than figment of his overactive imagination. So think of the play within a play as Hamlet's rational, university-educated, scientific mind trying very hard to come to grips with something his senses tell him cannot be true, but nonetheless is true, which is that ghosts exist, and that they are capable of speech.

So The Mousetrap functions as a test for Hamlet. He will determine, from Claudius' and Gertrude's responses, whether or not his father's ghost is telling the truth; he will also determine for himself whether or not he has the proof he needs to enact revenge upon Claudius as his father has commanded him to do.

But The Mousetrap is also a threat to Claudius, because it reveals that Hamlet knows, and it makes Hamlet a risk, in the same way that a murderer who is found out in a murder mystery, must be silenced. So there is a deeper theme of silencing going on in "Hamlet," the need to keep secrets, and never reveal too much of what is really going on--in the family, between Hamlet and Ophelia, and the ultimate secret of who killed the King.