to go out: to move from inside a place to outside it, or to come forth from a source into view
This is the plain, everyday meaning of the word: leaving an enclosed or interior place for the open. A person walks out of a house, a city, or a tent; an army marches out to fight; a child comes forth from the womb. The same picture stretches naturally to anything that emerges from a source. The sun comes up over the horizon, water flows out of a spring, a shoot springs from a stump, a boundary line runs out from a fixed point across the land, and words come out of a mouth.
What holds all of these together is the direction of the movement, not the thing that moves. Something that was contained, hidden, or fixed in place now passes to the outside and becomes visible or active. Hebrew uses this one verb where English reaches for a whole shelf of words: go out, come out, come forth, depart, proceed, rise.
Because the focus is on crossing from inside to outside, the word is the natural opposite of בּוֹא, the verb for going in or coming in, and the two are often paired to sum up the whole of a person's activity, their going out and coming in.
The law also gives the verb one fixed legal turn: a bound servant who has finished his term is said to "go out," meaning to leave service as a free person.