Some explanations I haven't seen here...
I'm posting this because as far as I can see, none of the threads touch on the things I've seen in the film.
First, regarding the beginning and ending of the film. There's a thread here about the differences between the two scenes, and I was surprised that no mention was made of the fact that the songs Llewyn sings are different in each scene, so it can't be a flashback. There is, in other words, a significant difference between the two scenes. Why does the film begin and end the same way? Jean tells him, "You don't wanna go anywhere, and that's why all the same *beep* is going to keep happening to you. Because you want it to."
The songs are also a clue to film's meaning. For example, "The Death of Queen Jane" is about a dangerous pregnancy, with Queen Jane asking to be cut open so that the baby will be saved. In the end, she dies giving birth to it. This is not a direct parallel to the plot of the film, but it must be poignant for Llewyn who has just learned that an abortion he paid for two years ago was never undertaken and that he is in all likelihood a father. (It occurs to me now as I write this that this may be another example of an eternal recurrence. In two years he could be trying to pay for another abortion, learning that the previous one again was not undertaken.)
One thing that connects all the songs that are sung in full in the film is the word "farewell":
Although "Hang Me" doesn't feature the word, the singer is singing about his death, saying that he wouldn't mind it.
"Dink's Song" (which Llewyn had sung with Mike) is also called "Fare Thee Well".
"The Last Thing On My Mind" is about going away. "Are you going away with no word of farewell? Will there be not a trace left behind?"
"Five Hundred Miles" -- another song about leaving.
And when Dylan appears, he sings "Farewell".
The last word of the film? "Au revoir." (This could be significant, because it means "Till we see each other again" -- a recognition of the film's looping structure.)
And this leads me to the last point. We know throughout the film that Llewyn's partner Mike killed himself, although we never learn why. When he auditions for Bud Grossman, Llewyn says, "I had a partner." And Grossman says, "My suggestion? Get back together." Grossman doesn't know Mike, so he doesn't realise what he's saying. But Llewyn says, "That's good advice. Thank you Mr. Grossman." I think the very least we can gather from this is that Llewyn is considering joining Mike and killing himself.