This first episode is actually the movie that the SOS Brigade makes for the school festival. (The story of the making of the film doesn't appear in the first season, although it does exist in the novels.)
The movie features Mikuru Asahina as a mild-mannered time-traveling waitress who is in place to try and prevent Yuki Nagato, an alien witch, from destroying the Earth. Mikuru is also supposed to protect a man named Itsuki Koizumi, who is important to the future. She ends up falling in love with him, as does Yuki, and the film devolves into a love triangle comedy.
The episode (and, in fact, the entire film?) is narrated by Kyon. The director is Haruhi. Neither of them appear in the film itself, although they do appear briefly at the end of the episode.