To really want something very much.
I'm dying for a cold drink on this hot day.
이렇게 더운 날에는 차가운 음료를 마시고 싶어 죽겠어요.
To really want to do something.
She's dying to tell everyone the good news.
그녀는 좋은 소식을 모두에게 말하고 싶어 죽겠어요.