Why might Shakespeare have framed up his story in this way?

The Taming of the Shrew opens with a framing story, labeled the Induction in the text. This sort of device was quite common during the Elizabethan era. Nonetheless, it is worth noting the connotations of the word "induction" - as if we the audience were being inducted into a ceremony or institution in our honor. Sly is led to believe as much, falsely "inducted" as he is into the nobility. The entire play thus emerges as a device to fool the drunkard - and, by extension, us. The Lord is thus a representation of Shakespeare himself, staging a set of carefully controlled and convincing illusions. However, whereas the typical theater audience succumbs merely to the illusion of the stage, Sly succumbs to illusions about his own self. He must submit to the new identity the Lord has fashioned for him. In other words, he must become not merely spectator but an actor and character.