Teaching a 150+ student lecture, it's quite hard for a professor to understand what the "common problems" are in a classroom. Not all students have the time to talk with the professor after class, since many have other classes to go to, or perhaps a job.
A "clicker" (which I'll now refer to as PRS (Personal Response System), as that's what I'm used to) is a great way for a professor to ask a tricky question, and find out what the class thinks as a whole. As an example, my physics teacher asked a question about the direction of an acceleration vector. The question was, "If a ball is thrown into the air, which way is the acceleration vector pointing just before the ball is caught?"
Obviously, the vector is pointing down, but many students were confused about this, and answered either up or that it was the zero vector. This gave the professor a chance to address those students who answered wrong, and explain to them again why the vector was pointing down. Normally, by just raising hands, the professor could never get such an accurate answer of who thought what. He would then assume everyone understood completely, and just continue teaching, while half the class thought that the acceleration vector pointed up.
I've used PRS in quite a few classes. Many times, there is no grade on your actual answer to "PRS quizzes", but rather just a "participation grade", which further translates into just an attendance grade. Many professors don't know how to properly use the system to the advantage of themselves or the students. On the other hand, some know very good ways in which to make a possibly boring giant lecture hall into an interactive room with actual learning taking place.