To understand the meaning of lines 44 through 51 of Act 1, Scene 1, it can actually help to look at lines just before that to become better oriented to the situation. In lines 41 through 45, Sampson and Gregory, two servants of the house of Capulet, have just started an argument with two Montague servants.
Prior, Sampson and Gregory were debating whether or not Sampson would actually have enough nerve to challenge Montagues. Now that Montagues have appeared in the same public place, Sampson starts a fight with them by "bit[ing] [his] thumb at them," which is an insult, the equivalent of flipping someone off (I.i.38). However, Sampson wants to be sure that the Montagues can be legally held responsible for starting the fight, which is why he denies biting his thumb at them, and rather says that he is just biting his thumb, as we see in his lines, "No, sir, I do not bite my thumb at you, sir; but I bite my thumb, sir" (44-45). Hence, lines 44-45 simply mean that Sampson is denying he has insulted them, just because he wants them to become angered enough to start the fight on their own.
Next, Gregory asks Abram, Montague's servant, if he is trying to pick a fight, which Abram denies in line 48. Then, Sampson tries to encourage the fight even further by saying that if Abram is trying to pick a fight, then Sampson is as good a fighter as Abram, as we see in the lines, "But if you do, sir, I am for you. I serve as good a man as you" (49-50). Line 51 ends with Abram intentionally insulting Sampson, thereby encouraging the fight further, by asking Sampson if he only fights as well as Abram does and not any better, which is how we can interpret the line, "No better," meaning, "No better [than me]?: (51; eNotes).
Hence, lines 44 through 51 are merely an example of boys being boys and trying to pick a fight with each other, due to the two households' longstanding feud.