What does the violent argument between Hamlet and Laertes add to the play?
Hamlet's disruptive behavior at the funeral of Ophelia is uncalled for, except by the standards of the rebellion against society he has chosen to follow. He does not want to allow Laertes to mourn for Ophelia. Obviously it is Laertes' oblique reference to Hamlet that triggers Hamlet's outburst:
Oh treble woe
Fall ten times treble on that cursed head,
Whose wicked deed thy most ingenious sense
Deprived thee of !
Hamlet's reaction is a microcosm of the way he acts throughout the entire play. He disregards others' feelings, but speaks with such eloquence that he seems to be expressing universal truths that the audience instantly can identify with:
What is he whose grief bears such an emphasis,
Whose phrase of sorrow
Conjures the wandering stars, and makes them stand
Like wonder-wounded hearers?
This is I, Hamlet the Dane!
As unfair as it is of him to act as if Laertes has no right to express his grief, we have to recognize that Hamlet's side is valid in some way. The sense I get of Laertes is that, grief-stricken or not, he is basically false and deluded like the other courtiers. He had "guilted" Ophelia in his moral lecturing of her, and this could indeed have been one factor leading to her suicide. Regardless of who is right or wrong in the graveyard scene, the fight brings matters to a head. From this point there is no turning back, and the king can now use Laertes in his plot to destroy Hamlet. The scene also shows, in stark terms, the tragic effect of Hamlet's behavior upon his mother. "Oh, my son, what theme?" Gertrude pathetically cries when Hamlet says he will fight with Laertes upon this theme "until my eyelids will no longer wag." As his mother she is the only one among what we might call these "establishment" people at court who has any empathy for him.
The violent clash between Hamlet and Laertes in Act V shows a lot about both of their characters. It shows that Hamlet wants to do the right thing (apologize), and underscores his affection for Ophelia. It shows that Laertes wants to do the right thing—but that he is as committed to his revenge for Ophelia and Polonius as Hamlet is to his revenge for his father. It shows that both can be manipulated (explicitly, by kings and father figures), that both share family feeling, and that both can be driven by events larger than they are. It also adds drama and literal violence by ending in a duel.