In the opening scene of the play, two Capulet servants, Sampson and Gregory, attempt to start a fight with a servant from the house of Montague as he is walking past them. They are aware of the law prohibiting fighting among the Capulet and Montague's and decide to insult the Montague servant by biting their thumb at him, which was considered extremely disrespectful in the Elizabethan era. The servants then begin to fight as Benvolio enters the scene. When Benvolio attempts to break up the fight, Tybalt enters the scene and attempts to fight Benvolio. Benvolio and Tybalt then begin to fight, and a melee ensues on the streets of Verona between the two rival houses. Even Lord Capulet and Montague enter the fight. Eventually, Prince Escalus enters the scene and breaks up the fight. The melee that takes place in Act One, Scene 1 begins when the servants from both houses begin to fight each other. 

The fight is a mere flare up in the feud between the Montagues and Capulets. And in reality, this particular fight is really among the servants of the two houses. (Even Benvolio is little more than a two-bit role player in the ranks and file of the House of Montgue.) However, when Tybalt enters everything changes. Now on the scene you have one of the leading members of the family, one notorious for his short fuse and his pit-bull like distemper nonetheless (Baz Lurhman's film captures this element beautifully.)With Tybalt involved, what was a rather minor fight within the feud is catapulted to another level that requires severe action from the Prince.

What really causes the fight is the fact that the Montagues and Capulets hate each other.  They hate each other so much that even their servants hate one another.

It is these servants who are the immediate cause of the fight.  Sampson and Gregory (Capulet servants) are walking along when some Montague servant come out.  Sampson insults them and they start fighting after a bit of a verbal argument.

When Tybalt comes in, he makes the fight worse.  Benvolio has been trying to break the fight up.  But Tybalt decides that he wants to fight Benvolio because he hates Montagues so much.