This idiom or expression is difficult to explain. Please look at the following examples:

When someone says: I’m going on a trip, eat your heart out, they mean something like:
I’m going on a trip but you’re not. I hope that makes you jealous!

When someone says: I just got a promotion, eat your heart out, they mean something like:
I got a promotion but you didn’t, and I know that’s killing you!

Depending on who says it and how it is said, this could be a mean or a humorous statement.