Tybalt and Mercutio begin arguing in this scene almost from the moment Tybalt arrives. When he begs "a word" of Mercutio, Mercutio immediately sneers, "Couple it with something: make it a word and a blow." They continue to argue for a short time before it begins to escalate, at which point Benvolio, as perhaps the least hot-tempered member of the group, intervenes.
He attempts to prevail upon their sense of honor and, perhaps, their self-consciousness to get them to stop arguing by pointing out that "we talk here in the public haunt of men." He suggests that they should either "withdraw unto some private place" where they can continue arguing without being watched by the gathered masses, "or else depart." He is telling the two men, essentially, that they are creating a public spectacle of themselves and that if they must continue arguing, they should go and do it somewhere where they won't be creating a scene.
When Romeo enters, Tybalt says, "here comes my man," indicating that he has been waiting for Romeo—it was Romeo he was hoping to find in coming here. He tells Romeo that he hates him so much he can describe him in no better terms than saying "thou art a villain."
Romeo quietly declines to fight Tybalt, for reasons which he cryptically explains. He says that he has "reason...to love" Tybalt but will not say what it is. He then says that the name Capulet is one he now holds "as dearly as my own." The audience knows from Romeo's words, then, that he does not want to fight Tybalt because, as far as Romeo is concerned, there is now a familial connection between the two of them through Juliet. However, he does not actually express this to Tybalt because his relationship with Juliet is still a secret. The result is that Tybalt does not understand why Romeo is behaving this way, and Mercutio calls it "vile submission."