I just rewatched Moulin Rouge, and to be clear, here is the frame for the story.
On screen, we see a red curtain open up and images start to flicker on a screen; this is frame one, an audience watching a film.
In the image, we see Toulouse singing about Christian; this is frame one, Toulouse singing about the story.
We then flash to Christian, writing on his typewriter; this is frame three, Christian writing his story.
We're at least three steps removed from the actual story: we're watching a movie about a movie about a guy telling the story about a guy telling a story. There's a sketch in The Kids in the Hall about two guys sitting in a bar talking about a movie one of the guys watched the night before, and then we see the movie, and in the movie, two characters are talking about a movie one saw where two guys were sitting in a bar discussing a movie. There's an episode of The Simpsons in which characters share stories and read notes that bring us into stories within stories within stories to an absurd degree. That's what this seems like, but it's for real. And then, of course, Christian's story is about writing a play that reflected his own life story. I'm getting tired.
Other famous frame stories? There's Heart of Darkness, where a guy tells the story about hearing Marlowe tell a story. There's The Turn of the Screw, which if I remember correctly, a guy tells a story about another guy telling a story he heard (there may have been somebody writing the story down). There's The Rime of the Ancient Mariner, a very famous frame story. There's Frankenstein, where Frankenstein tells his story to a particular ship captain. Of course Don Quixote may have started this all: the narrator describes reading and translating from a particular chronicler or Quixote's adventures. The frame story is a very old function and convention of Western literature.
The benefits of a frame story? Sometimes the frame itself can add thematic meaning. Sometimes it places the audience into the story. Sometimes it renders the narrative unreliable.
Think about frame stories.