Of course there is a narrative, no composer ever came closer to making a piano sonata into a song.
I could talk about the tonal stasis of the first and second movements, but that would take hours to get into even with the slightest detail. Instead I'll make a general point of interest - the unforced and forced changes of direction. In any sonata form composers curb an improvisational attitude to harmonic change, most famously in the Liszt sonata. The transition to the second subject of the first movement is particularly forced, while the opening up of dimension to the famous middle section of the slow movement is fresh and unforced. The combination of these aspects throughout the work give a push and pull effect on the listener - while retaining the dream-scape that Schubert is so famous for. People always talk about how Schubert stops time - but they forget the rootedness of his structures, and his unerring compromise on behalf of them - in contrast to, for example, Schumann(Fantasy in C first movement).
The first and second movements contain some of the most sublime music ever written. The third and fourth disputes the quality of the Sonata. The baseness of the final movement bothers me - but we always have the great D960 to ponder.