How does Romeo find out that Juliet shares his love?
Romeo hears Juliet confessing her feelings when he is standing in her garden and she's speaking to herself on her balcony.
After Romeo sneaks into the Capulets' ball and sees Juliet, he finds himself instantly in love with her. As the Chorus details in the prologue of act 2, he's set aside his love for Rosalind in favor of Juliet. He doesn't want to be parted from her. Once he's under her balcony, he sees her approach and listens to her speech.
Juliet wonders out loud why Romeo has to be a Montague, the sworn enemy of her family. She says that she wishes he could forsake his family and that she'd forsake hers for him if he declared his love. Juliet says that she wants him to forsake his name to be with her.
Romeo—whose presence was unknown until this point—answers and offers to do so.