Quite simply, Romeo overhears Juliet proclaiming her very famous "What's in a name?" speech and, in doing so, declaring her love for Romeo. Ironically, Juliet's speech stems from the age-old feud between the Montagues and Cauplets. Juliet says, "Deny thy father and refuse thy name! / Or if thou wilt not, be but sworn my love, / And I'll no longer be a Capulet." This, in a nutshell, is what Juliet is saying here: refuse your name of Montague, Romeo, or I will refuse my name of Capulet; in this way, we can become lovers. In regards to Juliet's feelings for Romeo, although she doesn't use the word "love" here, she does say, "Take all myself." Of course, perhaps she would have uttered the word "love," but Romeo interrupts her before she can do so.
In the famous balcony scene of Act II, Scene 2, Romeo overhears Juliet express her love for him and her anxiety about who he is.
On the balcony, Juliet sighs, "Aye, me!", indicating that she has been pondering something that troubles her (2.2.24). Then, Romeo overhears her mention his family name, a name that is one of enmity to her family. Just as Romeo wonders whether he should speak up, Juliet declares that it is only Romeo's being a Montague that is an impediment to her love because were he not a Montague, he would be the same person:
'Tis but thy name that is my enemy.
Thou art thyself though not a Montague.
What's Montague? It is nor hand, nor foot,
Not arm, nor face, nor any other part
Belonging to a man. (2.2.38-42)
Juliet adds that she wishes Romeo could trade in his name and take all of her in exchange. When she makes this entreaty to Romeo, he impulsively responds,
Call me but love, and I'll be new baptized.
Henceforth I never will be Romeo. (2.2.50-51)
At the sound of his voice, Juliet is startled and embarrassed that he has overheard her. She asks Romeo how long he has been standing below her balcony, and she wonders how he has gained entry to the garden below. Furthermore, she fears that if any of her kinsmen see him, he will be killed; however, the enamored Romeo is not worried. Juliet then admits that she would not have declared so easily her love had she been aware of his presence. Further, she asks Romeo to not assume that since he has made her love him that her love is not serious.