Why do professors ask candidates at PhD interviews what specific problems they are interested in?
Other than the previous answer, we ask this question because a candidate may be interested in an area that is not in fact well represented in the department. If someone wanted to do research on numerical methods for optimization, my department has nobody to advise such a thesis -- but the candidate may not be aware of this.

I personally ask this question because too many graduate students come to me and ask whether I could be their adviser and then expect me to pose a problem they should work on for the 3-4 years. But I don't want to do that: if a student doesn't know what they want to do for research, yes, sure, I can give them a topic, but if it doesn't match their interest, it's going to be a painful 3-4 years for everyone involved. So I typically want that my students come up with topics they find interesting themselves, and then we can see how that fits into my research group and program.