Hello @jonasvestergaard, here is my understanding...
There is a way in Spanish to say that one "has to" do something. The structure of this is: (conjugated form of tener) + que + verb ( in other words, that which one "has to" do)
Hence, the part of the phrase "tenemos que llegar" means, "we have to arrive," or, in this case, they have to be at the airport at six. If "llegamos" was placed in the spot "llegar" holds, the translation would be: "Because we have to we arrive at the airport at six." This is confusing at first because you could easily say, "Porque llegamos al aeropuerto a las sies," which means, "Because we arrive at the airport at six." This will get the message across, but the speaker is adding a tad of urgency to this bit of the conversation and it sounds more pressing to say something like "we have to (tenemos que...), we ought to (debemos que...), or we need to (necesitamos que...)."
For the second part, I would say puerta de salida basically means "exit door" since salir is the verb for "to leave." Direct translation would yield "door of departure" or something to that effect. So in this case, "salida" isn't a conjugation of salir, but a word used a lot like "safety" would be used in the phrase "safety exit" (which, by the way, "safety" in Spanish is seguridad).
The vosotros form is used in Spain all the time as the plural form of "you," although it isn't used in Central America. Thus, it is put in the course for memorization because a lot of individuals learning Spanish will travel not just to the South American countries but to Spain as well. I've never thought that learning the extra conjugations for vosotros was any fun, but I've just accepted it as a necessary evil if I expect to be fluent in both the Americas as well as overseas.
I hope this helps:)!!