You know, home is the ultimate place where each of us should normally be. So in Korean, "go home" is often expressed as 집에 들어가다 ("집에 들어가면 가족이 있다"). 먼저 들어가겠습니다 is essentially a shortening of 먼저 집에 들어가겠습니다. People just say 먼저 들어가겠습니다 because it is so obvious that one is going home if he doesn't say anything special. We just say 들어가, 들어가세요, 들어갈게요, etc. as a parting word, especially at night because everyone is expected to go home at the end of the day. I was surprised to see this common usage not listed in the dictionary, but it is very widely used, at workplace, on the phone, parting ways on the street after dinner and such.