This idiom or expression like phrase means:

Very jealous.

You should have seen her looking at my dress. She was green with envy, maybe even greener than my dress!!!

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.