The art history professor says that Costa criticizes his theories about the distinction between different types of paintings based on the belief that "there are no features possessed by all and only the works from a given historical period" so "assigning works of art to a period style is intellectually bankrupt." In other words, there is no point to categorizing artwork by period because there are no unique characteristics owned by all of the artwork in each category exclusively. However, Costa has his own theories on transitioning from different periods of opera, and these theories require the same type of period classification as Costa criticized. Therefore, the art history professor argues that Costa's reasoning is hypocritical and can be discounted.
This is an ad hominen (personal) argument. Instead of addressing Costa's argument on its substance, the art history professor discounts it because he sees Costa as hypocritical. This is irrelevant to the substance of the claim/not reason enough to determine that an argument can be discounted, and answer choice (C) points this out. It reads "the argument rejects the reasoning on which a criticism is based merely on the grounds that that very criticism could be applied to theories of the person who offered it." This identifies the flaw and is therefore correct.
Does this make sense? Please let us know if you have any additional questions or would like an explanation as to why the other answer choices are incorrect.