The plot in act 1 scene 1

baptisia has daughter,  Bianca, Katherina. Katherina has to find a suitor and marry before Bianca. But Bianca has already got a man, Hortetensio. lucentio and Tranio want to continue in the education. Lucentio wants to marry Bianca but she is not interested in him so they swap clothes, him and Tranio. He now looking like Tranio the tutor for Bianca, this is why they swap. If he is her tutor her can spend time with her and get closed to her.