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’ll make you crazy!”

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.