Why does the play begin wih a fight scene?
Not only does the ensuing fight establish the conflict between the Capulets and Montagues but this opening scene follows the same pattern of all of Shakespeare's plays.
Shakespeare's audiences were a mix of the educated members of nobility (who sat is seats) and the uneducated commoners who stood on the ground, known as the groundlings. The groundlings posed a serious problem to Shakespeare and any playwrite; they must be immediately hooked into the play and entertained or they would disrupt and ruin any performance. When we look at any of Shakespeare's plays we see that he addresses this situation by including one or more of three elements at the beginning of each of his plays: violence, references to sex, or references to witchcraft. Romeo and Juliet is no exception. In the opening scene we have violence and almost gratuitous references to sex--all designed to grab and hold the attention of the groundlings.
I think that it is useful to Shakespeare to start the play this way because it really sets up the atmosphere in which Romeo and Juliet are going to fall in love. By having this fight, Shakespeare can show just how much the Capulets and the Montagues hate each other.
In addition, he can show us how angry the Prince is about this feud. This will, of course, be very important later on in the play. Romeo will be banished because of the fact that the Prince is tired of this feud.
So, to me, the fight is just a very effective way to show us a couple really important things -- the extent of the families' hate and the fact that the Prince is tired of the feud.