The movie makes a number of changes to the plot of the book obviously, including significantly increasing the age difference between the two central characters, but the main plot point is the same: an orphaned girl is sponsored by an unknown benefactor to attend college. While there, the girl- Julie (not Judy, as in the book)- is lonely and starts to write letters to her mysterious sponsor though he never responds, which is what's occurring in the scene pictured.
I don't particularly like this movie, which is not one of Fred Astaire's better outings... I'm not really fond of any of the dance sequences and there's only one memorable song: Something's Gotta Give. To be fair, I'm not a huge fan of the novel either; I actually think that it's sequel Dear Enemy is the superior book. Leslie Caron is charming and obviously an excellent dancer, and Fred Astaire... well, he's Fred Astaire- what else needs to be said? But the movie as a whole doesn't do anything for me. Also, dance dream sequences are terrible in every movie... didn't work in Singin' In The Rain and definitely doesn't work in this film either.