The chief purpose of this touching family scene in Act I scene 3 seems to be the way in which is presents Hamlet and Laertes as foils or important opposites. This is of course something that is developed throughout the play, as Hamlet's procrastination and delay in revenging his father's murder is compared to the haste and desire of Laertes to slay Hamlet as swiftly as possible. Let us remember that immediately before this scene, Act I scene 2 presents us with a dysfunctional family and Hamlet's estranged state. His uncle, Claudius and his mother, Gertrude, both try to give Hamlet advice on putting aside his mourning and enjoying the festivities of their marriage, but this scene clearly presents a somewhat troubled family dynamic.
It is thus important to note the touching and normal family scene in Act I scene 3 where Polonius and Laertes appear to have a good relationship from what we can infer, and where Polonius offers his son bits of very sensible advice. Juxtaposing the two scenes together thus reinforces the way in which Hamlet's family is dysfunctional and his own sense of estrangement compared to the position of Laertes.