It means you feel so sad and miserable,
You could say that if you got dumped by a guy whom you really really loved
If you were asking the meaning of the phrase, It means same as 'black burning heart', literally 'heart which burnt black'
Example :
오늘 5년 넘게 사귄 남자친구로부터 이별을 통보받았습니다.
그로부터 헤어지자는 말을 듣자마자 새까맣게 타버린 내 마음을 그는 알까요?
Do you understand the meaning of the sentences ?
Hope this helped,
Lucky