It's an idiom referencing grapes. A grape, left unpicked, will sit in place and wither away if left too long. It basically means, "You want to know so badly that it's killing you".

"To be dying to...(verb)" means the person desperately wants to do that action. Adding "on the vine" creates a new analogy to exaggerate the idiom. The full phrase is a creation of the writer, so if you tried using it, you'd sound a little too clever for your own good.