1. To feel great sadness. I feel just awful for Mary—she's been eating her heart out ever since she found out she was rejected by her top-choice school.

2. To be very jealous. In this usage, the phrase is often said as an imperative and sometimes mentions a famous person (when the speaker comically claims to be more talented than that person). Eat your heart out—I got tickets to the concert and you didn't! Look at how well I dance now—Gene Kelly, eat your heart out!