Juliet is saying that names shouldn't matter. "A rose," she says, "by any other name would smell as sweet." But of course she knows that in her world names do matter. The Capulets are locked in a bitter old feud with the Montagues, a feud in which people hate each other only because of their names. Hence she says "Wherefore art thou Romeo?," asking essentially, "Why did you have to be born a Montague?" But ultimately, she realizes that she loves Romeo despite his name, and wishes out loud that he will deny his name, or that she could dey hers. Once Romeo joins her at the end of her speech, the audience sees that he agrees with her. Essentially, the lovers express their belief that their love will overcome the hostility that plagues the two houses. This, of course, as the audience already suspects, cannot happen, because neither of them can deny their names without consequence.