Why are characters fighting in Act 1?

The fight scene in act 1, scene 1 immediately introduces us to the background of the play. The bitter feud between the two warring families, the Montagues and the Capulets, has broken out on the streets of Verona in an unseemly brawl. Sampson and Gregory, two Capulet servants, are strolling along, looking for ways to provoke the hated Montagues. They come across two Montague servants and subject them to an insulting gesture. After a brief exchange of verbal insults, a fight breaks out, which soon escalates as Benvolio and Tybalt intervene on behalf of the Montague and Capulet sides respectively.
The proximate cause of the fight is the need to defend masculine honor. The Montague servants cannot simply walk away after being insulted by Sampson and Gregory; they must defend the honor of the family they represent by any means necessary. It would simply not be manly to do otherwise.
One of the many important aspects of the opening fight scene is that it shows us how deeply the conflict between the Montagues and the Capulets reaches down into Veronese society. This is a quarrel that involves everyone, from the highest to the lowest. The prevailing sense of masculinity displayed so starkly in the opening scene transcends the boundaries of class, making the problem of decisively ending the feud all the more difficult.