[A]느라 [B] B happened because (I was) busy with A (urgent) 급하게 나가느라 방청소를 못 했다. I wasn't able to clean the room because I had to leave in a hurry.

It means because/due to/since/ something . I can't answer your phone because I were driving my car 운전하느라/운전하는 중이였기 때문에 네 전화를 못받았다. I stayed up all night because I played a game. 게임하느라 밤새 깨어있었다. A느라 B했다. B is the result of A. You can add verbs just in front of A. 먹느라 = 먹다 + 느라 because of eating 자느라 = 자다 + 느라 because of sleeping 하느라 = 하다 + 느라 because of doing something