There are a couple of reasons why the Capulet servants decide to pick the fight with the Montague servants in the first scene.
One reason is that Sampson is trying to prove to Gregory that he is not a coward. This explains why Sampson says in the opening line that they will not "carry coals," meaning that they will not be humiliated and that they will "draw," meaning that if they see a Montague, they will draw their swords. Gregory, however, continues to call Sampson a coward saying that he is slow to use his sword in the line, "But thou art not quickly moved to strike," and that if he is moved at all, it is only to run away, like a coward.
The second reason why Sampson and Gregory pick the fight with the Montague servants is out of duty to their master, Lord Capulet. Since they know that Lord Capulet is having a feud with Lord Montague, they know that the families' servants are fighting as well. This is seen in Gregory's line, "The quarrel is between our masters and us their men."