To go out and away; to leave a place (e.g., a house or room). In colloquial speech the 'い' can be dropped (e.g., 出てった); a literary form may use 出てゆく.
彼は昨夜急に荷物をまとめて家を出て行った。
He suddenly packed his things and left the house last night.
雨が降りそうだから、早く店を出て行こう。
It looks like it's going to rain, so let's leave the shop quickly.
怒って彼女は部屋を出てった（出て行った）。
Angry, she stormed out of the room.
列車は駅を出てゆくと、次の停車駅へ向かった。
When the train left the station, it headed for the next stop.