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.