The usual reason is to collect information about how well (or poorly) your customer service operation is performing.
But another reason can be just as important: to leave the customer with the impression that the company cares about its customer service. That reason may never be articulated, but it can be just as important as the data collection reason.
If you just want to collect data, then all the research and rules of thumb about surveying technique apply. Be consistent in how you collect data. Be careful about how you word survey questions. Avoid sample bias. And so forth.
On the other hand, if the goal is to impress the customer, then other factors become more important. You probably want to use live interviewers rather than an automated survey, for example (and you'll want to select interviewers with good phone voices). Few customers get good vibes from a machine that plays "20 Questions," but a warm and friendly interviewer can leave a lasting impression. Quick follow-up is also important, since getting back to customers within just a few minutes shows that you're making a real effort to hustle.
With Express Feedback (which has been live for about two weeks at our first pilot site), we have a live human call a customer back shortly after the customer hangs up from our client's call center. Most of the time we reach the customer within 15 minutes. When we started Express Feedback, we were thinking that it was mostly about collecting better survey data faster.
But we've discovered something: customers are really impressed when we survey them so quickly. Our client looks like it really cares about the customer service experience (they really do care, by the way). There have been several interviews where the customer began the interview upset about something, but by the end of the interview was actually happy. Not happy because the problem was solved (it wasn't), but happy because the interviewer listened. It turns out that that's all most people want: to be listened to, and to feel like their opinions matter.