In a rather amusing section, Hortensio leaves the stage to teach Katharina the lute, and we are told that he returns shortly with "his head broke." Baptista comments on how pale he looks and then uses the animal imagery that you refer to in reference to his teaching of Katharina:
Why, then thou canst not break her to the lute?
To "break" an animal means to be able to subdue and dominate it, then train it, and so Baptista is asking if Hortensio will be able to train Katharina to play the lute, but the implication of the animal imagery is that Katharina is an animal that needs to be "broken" before she can learn anything--a very apt metaphor that results in Petruchio stepping up to meet the challenge.
In Act Two, Scene One of The Taming of the Shrew, the gentlemen (Gremio, Lucentio, Pretruchio, Hortensio, Tranio, and Biondello) arrive. Petruchio wants to see Katherine, offering her father, Baptista, music instruction from "Litio" (really just Hortensio dressed as a schoolmaster) in exchange for the visit. Baptista agrees, and Hortensio disappears to go teach Katherine how to play the flute.
When Hortensio returns, he is pale and has a bleeding cut across his head. Assured that Katherine would make a better soldier than a musician Baptista asks, "Why, then thou canst not break her to the lute?" Hortensio spits back, "Why, no, for she hath broke the lute to me."
The literal meaning of "break" here is "to teach"; Baptista is asking Hortensio if he cannot teach Katherine how to play the lute, and Hortensio is insisting that Katherine could teach him a thing or two. This is also a reference to the "breaking" of a horse--a critical part of training in which a horse is made manageable by humans via being taught to perform certain repeated behaviors. Baptista is insinuating that Hortensio must go to greater, more strenuous lengths in order to shape Katherine into a manageable pupil.