You are right to focus on the intriguing change expressed in the character of Katharina in this final scene, in comparison with her shrewish behaviour as expressed at the beginning of the play. The scene you make reference to is of course another public display of Petruchio's mastery over his shrewish wife, as he, in public, tells his wife that her hat does not "become" her and to throw it "underfoot." Her swift obedience, followed by her speech to the other wives on wifely servitude, given as instructed by her husband, clearly points towards the complete oppression and mastery of her husband over her. Note what Petruchio says before Kate enters, talking about what the obedience of Katharina promises:
Marry, peace it bodes, and love, and quiet life,
An awful rule, and right supremacy,
And, to be short, what not that's sweet and happy.
The happy ending won through Kate's obedience is only thanks to an "awful rule" and a "right supremacy" according to her husband. However, critics remain sharply divided about the meaning of such an ending. Are we really to believe that Shakespeare feels a happy ending is only possible when the wife completely becomes subservient to her husband and loses her very will? Either way, Katharina's display of obedience is enough to win Petruchio the wager, and they leave to enjoy their overtly happy ending.