누구시길래 is composed of 누구+시+길래 누구 means who 시 is an honorific particle 길래 is a colloquial form of 기에 기에 is a suffix of meaning cause. 원인이나 근거를 나타내는 연결 어미. 누구세요 means "Who are you?" 누구시길래 means "who the heck are you dare (do something)?"

당신은 누구시-길래/누구시-기에/누구신데 내 마음(-을) 애태우나요? Who are you to make my mind fret over? (Well, this is a rhetorical question, as you know.)