The correct answer is: A. talking nonsense.
The phrase “through his hat” is an idiom that means “to talk nonsense.” It is thought to have originated in the 16th century, when people would sometimes wear hats with holes in them. If someone was talking nonsense, it was said that they were “talking through their hat.”
The other options are incorrect because they do not accurately describe the meaning of the phrase “through his hat.” Option B, “talking ignorantly,” means to speak without knowledge or understanding. Option C, “talking irresponsibly,” means to speak without considering the consequences of one’s words. Option D, “talking insultingly,” means to speak in a way that is offensive or hurtful.