가시나 can mean two different things.
1. 가시나(n) - dialect for "a young girl". Even though a dialect word, it's still widely known.
2. 가시나(v) - 가시다(honorific form of 가다) + 나(a question form asking something vaguely, as if to oneself)
So the title is a play on word, as it can mean both someone leaving and the girl (the singer) herself.
두고 가다 is in the [V1 + -고 V2] form, with the verb 두다, to leave something somewhere/behind, and 가다.
This form means performing two actions in sequence(밥을 먹고 나갔다), but sometimes the two may be inseparably integrated(차를 타고 갔다).
두고 가다 is thus more like a single action with two aspects, of leaving someone behind(두다) and departing(가다).
왜 예쁜 날 두고 가시나 is something like "Why is he leaving behind a pretty one like me?".
Other examples:
- 밥을 먹고 나갔다 = 밥을 먹은 뒤 나갔다 - (I) ate a meal and went out.
- 아버지는 내 손을 쥐고 천천히 말씀하셨다 - Father held my hand in his and spoke slowly.
- 짐을 싣고 가는 트럭 - a truck transporting a cargo.
- 동생은 어디서 강아지 한 마리를 데리고 왔다 - My brother brought home a puppy he had gotten somewhere.