Shakespeare loved to use the theatre and theatrical conventions to create a mirror to real life. Of course, to a large degree, that is the point of the plays themselves -- to cause the audience to witness the events that take place on stage and then turn their gaze inward and see the same dilemmas confronted by the characters on stage mirrored in their own hearts. In Hamlet, this theatrical device is made literal when a group of traveling actors actually appear as characters in the play. They relate to a very common theme in Shakespeare's works -- the theme of appearance (or pretense/illusion) versus reality.
The most obvious way that this theme reflects the events of Denmark is that Hamlet creates a play for the actors to perform (illusion) that closely mirrors the events described to him by the ghost of his father (reality). He plans to have the players perform this "Mousetrap" in order to see if Claudius reacts in a way that confirms his guilt. In this way, Hamlet will use "appearance or illusion" to confirm "reality."
But the events of Denmark also includes Hamlet's grief and his debate over taking the action necessary to revenge his father's murder. When he sees the First Player, in Act II, scene ii, present the illusion of real grief -- "all his visage wann'd/Tears in his eyes" -- he chides himself that he, with a "real" reason for grief can do nothing.
Also, the speech delivered by the First Player is the retelling of the death of King Priam and the grief-stricken response to this of his queen, Hecuba. This "illusion," again, contrasts the "reality" of the situation in Denmark where, rather than rage in grief as Hecuba does, Queen Gertrude has married her husband's murderer.
The players play a significant part in demonstrating the theme of appearance versus reality in the play Hamlet, contrasting the emotional highs and overt displays of grief enacted by the player with the inaction of Hamlet and the un-widow-like behaviour of Gertrude with the story of Hecuba. It is, however, "The Mousetrap" that presents the most important connection to this theme, wherein Hamlet uses "appearance" to prove "reality."
I would agree with everything in the first post - it is full of good ideas and it's right!
There is ananother aspect: the players arrive from outside Elsinore - they are a breath of fresh air. The world of Elsinore, the life in the castle, is very claustrophobic and the players break through that and bring art and creativity into Elsinore. On a further practical note, becasue they are from outside Elsinore Hamlet knows he can trust them - they will not be spying on him like Rosencrantz and Guildenster. Later, when Hamlet gets the chnace to leave Elsinore and have adventures with the pirates, he returns renewed, refreshed: Elsinore is a prison and the players represent the outside world.
Professor Bloom chose for his little book the title, HAMLET POEM UNLIMITED, quoting Polonius(2.2.394, Oxford ed.). Someone(Harry Levin?) wrote at some length about the speech that Hamlet asks the player to recite. I'll go look for it. In the meantime, politics or political expediency, the nature and purpose of drama and, here recently broached as a topic, morality are some themes. The source of the players speech, as they say, is the work of Virgil, who lived not long before the time of Jesus. The last words of the scene, "The play's the thing / Wherein I'll catch the conscience of the King"(2.2.594), might recall a comment from the Duke on the play within the play A MIDSUMMER NIGHT"S DREAM: "A very gentle beast, and of a good conscience"(MND5.1). Also, the line from Claudius, "How smart a lash that speech doth give my conscience"(3.1.51), soon follows.