She hopes Sita will reject her brother and then he will then kill her. She wants him to steal Sita, so that she can have Rama for herself.
In order to get revenge on Rama and Laksmana for their rejection and cutting off her ears and nose, Surpanakha goes to her brother, Ravana and describes Sita's beauty -which arouses his lust. Surpanakha then tells her borther, "She is a fit bride for you, and you are indeed the most suitable suitor for her.