What purpose is served by Hortensio/Licio's account of his music lesson with Kate?

As with many of Shakespeare's romantic comedies it is a case of people pretending to be other people. Both Hortensio and Lucentio are in love with the fair Bianca. The scene (Act 3 sc.1) serves to display the competition between these two men for the love of Bianca. It also shows that Bianca is smarter than the two men know; she plays her own game with them. Take a look at GradeSaver for more detail of that particular scene. I'll put it in the source link.

that doesnt help at all