Samson and Gregory fight with Montague’s men because they are Capulet servants and the two families are feuding.
A feud is a long-standing argument between families that can sometimes last for generations. In old Verona, that meant that everyone associated with one family had to hate the others. The two families fought each other on sight, and so did their servants.
The Montague and Capulet families are both important in Verona. Almost everyone has aligned with one or the other.
As Sampson and Gregory walk along, the feud is on their minds.
I strike quickly, being moved.(5)
But thou art not quickly moved to strike.
A dog of the house of Montague moves me. (Act 1, Scene 1)
Any insult can turn into a bloody duel. The men walk around with swords and are ready to use them. Gregory comments that the “quarrel is between our masters and us their men” and Sampson says it’s all the same. When they see servants of Montague pass by, they decide to provoke them, starting a brawl right there in the marketplace.
The play opens with this scene of senseless violence in order to provide exposition, and give the audience a taste of what life is like in Verona.