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.

I'm not familiar with CS departments in particular, but I think this is less a "we ask because we want to know what we can expect you to hit the ground running on" and more of a "we ask to see what you know about us and where your general interests are". It's pretty much a job interview, so take it as a job interview type of question. A company wants you to show an interest in their company that goes beyond simply submitting a job application. They want to know what you know about them, and how that influences your job hunting. People that simply throw in an application and demonstrate no knowledge or specific interest in the company are unlikely to be hired unless the position is rather non-competitive. Companies want interested, happy, inspired workers; not some guy just looking to cash in his next paycheck, and likely biding his time for something he'd enjoy more to come along.

Look into the department before you arrive and see what things the faculty are working on now, or have worked on recently, and single out ones that seem most interesting to you. If you find there are none, this probably is not the right university for you.

This isn't something that binds you in any way. As one of the qualifying exams for my Ph.D., I was to present a talk on some recent research to demonstrate my knowledge and interest in doing research. Ostensibly, interest specifically in the subject matter of the talk, but in no way was that binding. My actual thesis could, and ultimately did, have absolutely nothing to do with the subject matter of this oral exam. To this day I have not produced a single piece of research that's relevant to the topic of that talk.

A candidate with no intrinsic motivation and own research interest appears to be only in for the name of the lab and the title of PhD, not the topic.

You want creative students that are eager to pursue research and come up with ideas of their own. When you are reading the research topics of the lab, and you don't come up with ideas, then you are the wrong guy. It may be possible to choose the advisor later - but you really need to be highly motivated for the specific topic of at least one of them...

I'd never accept a candidate with a "tell me what to do now" attitude, or a "maybe I will find out later what I like" attitude. It's okay if you switch topics (if it's early enough) but you have to burn to do this.