This is a lesser-known work of Tchaikovsky's, written while he was on vacation in Beyreuth. He was coming off of a very long time of depression, and was inspired to write this when he read the story of Francesca and her lover Paolo in Dante's Inferno. Despite the distressing subject matter of this tone poem, this piece marks the return of Tchaikovsky's creative spirit after a time of lethargy.
The story goes like this. Dante and his guide, Virgil, have entered the circle of Hell reserved for people guilty of the sin of lust. They are punished by being thrown about in a whirlwind, similar to the 'whirlwind' of their passions, for all eternity. Dante sees two figures in the whirlwind that he feels very sorry for, and asks them to tell their story. It is Francesca, who was married to a cruel man she didn't love, and had an affair with his handsome brother, Paolo. Her husband found out, and murdered them.
It's a sad and somewhat disturbing story, which might explain why it's not a very popular piece of Tchaikovsky's. Oh yes, the Romeo and Juliet Fantasy-Overture ended tragically for the two lovers, but at least the ending chorale promises peace between the Montagues and Capulets. Francesca da Rimini's story and in turn, Tchaikovsky's tone poem, offers no such light. Except for maybe the light of Dante's compassion and pity for the couple. This might have been what appealed to Tchaikovsky about the story (aside from the themes of tragic love, which he seemed to like).
Musically, too, this isn't quite as engaging of say, Capriccio Italien or R+J, but it is still very good and interesting. The opening goes on a little too long, but it is very evocative of a chaotic, unrelenting whirlwind. It makes you feel kind of helpless, and like good writing, you feel very immersed into the setting and mood.
Then, after this chaotic and infernal opening, things quiet down suddenly as a bass clarinet plays a plaintive solo, representing Dante asking the lovers for their tale. We are suddenly whisked away to a totally different scene in Italy, as Francesca relates the story. The bass clarinet's plaintive song is developed and carried by different instruments, swelling with passion until the discovery of their affair and their subsequent murder. The music ends dismally- with the return of the whirlwind.
The subject of tragic lovers appealed to Tchaikovsky, as we've seen in Swan Lake and Romeo and Juliet before. This may not be his best work, but it's a poignant and haunting comeback for a composer with many brilliant works still ahead of him.