Earlier quoted context omitted.
Is "I go first" a correct translation? If so, why is it used? As an excuse that you have to leave earlier than the person you have been with? Or to avoid that uncomfortable situation where both parties aren't sure if it is time to leave, and are just standing around awkwardly?
When you see these things in foreign languages, it may seem odd. However, if you look at the origins of words and phrases in English, it is also full of odd expressions. It is just that you are used to them. Take "welcome."[1] Well come. You've arrived well. You make a good entrance? Not a ridiculous thing to say when someone arrives. What about in response to "Thank you?" You are welcome.[2] Huh? What does that mean…
The french "bienvenue" seems to be a too-literal translation from wiljakumo [0]. A similar construction is missing in Latin.