Actually, Sampson and Gregory are servants of the Montague house and actually fight servants from the Capulet house.
In Act I, scene i, Sampson and Gregory are talking up their loyalty to the Montague family. They are attempting to seem macho saying how they would take on any Capulet, even a dog belonging to the family. They go as far as saying they would rape the women of the house of Capulet.
They then see two servants from the Capulet house, Abraham and Balthasar, and Sampson tells Gregory to start a fight (line 27) Capulet men into starting the fight and get the law on their side so that they don't get in any trouble, so they start thumbing their noses at the Capulet men (kind of like flipping someone off today).
When the Capulet men ask if they are thumbing their noses at them, Sampson and Gregory deny it. However, they then see Benvolio, a kinsmen, or relative, of their Master - Montague - so, in trying to impress Benvolio, they end up starting the fight themselves saying that the Montagues are better than the Capulets.
Basically, they start the fight to prove their loyalty to the Montagues.