In a phone communication, we use "먼저 들어가겠습니다." to express "I'll hang up first." or "we've finished the conversation." (in many cases.) Even though you are at home(so you don't need to go home), we use this expression.

It means "enter home" "먼저 들어가 보겠습니다" means "Excuse me for leaving for home before you do"

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.

The situation : In the office, between coworkers. I'm leaving in advance : 먼저 들어가겠습니다. You can use this sentence when you leave for your home at the party or drinking with others.