According to Robert Cummings, the guy who wrote the blurb about the piece for www.allmusic.com, the duo was the last thing he wrote before leaving Bonn for Vienna in 1792. He wrote it for his "law-student friend, Degenhart, to whom he also dedicated the work. In a note on the autograph score, the composer declares the piece is a "souvenir" to mark his approaching departure to Vienna".
I hope his friend Degenhart played the flute. I also hope he had another friend who played flute, since Beethoven was leaving town. It's a bit like giving your friend a tandem bike with a note that says, "Enjoy. I'm moving."
Robert Cummings also wrote "Here is yet another work whose publication Beethoven suppressed throughout his life. It is likely that he would have strongly opposed its posthumous appearance, viewing this duo as an early effort meant for a friend, not the public."
I think Robert's right. And what Beethoven would've been trying to suppress, was his work getting published and played by professionals or amateurs to limited audiences. Imagine if he'd known that, one day, any individual could record themselves playing his music, at whatever skill level they possessed, and then put that recording on a platform viewable by the entire planet.
Like his just-for-my-friend flute duo, played by one guy, with himself:
And in case you want to hear the allegro too, here's the whole thing played by two people: