Use a rubber duck.
Put a little rubber duck near the user. Tell the user the rubber duck's name - the more non-fitting, the better. Mine is called Frank The Duck.
You see, Frank The Duck is a bit dumb. I tell the user that Frank doesn't know how to use the system, and so I need to teach it. The problem, however, is that since I'm a technical person, I'm really bad in explaining the things that need to be explained in practical, everyday terms - regular English.
So, now, the user has a little humorous mission to undergo with me. I'm putting the user on a position of power, making him teach someone, giving him the reins of the experience. I tell the user that I'm the incompetent one, so the user feel more confident about himself and what should be done. And, them, I incentive the user to explain stuff as he would teach someone.
Teaching is a interesting experience. It puts your brain on a different mode that changes a lot how you think and how you speak, creating some sort of direct bridge between your ideas and your mouth. If the user needs to explain, he will feel that what he needs to do is a bit pointless - you already know what he is doing, why should he bother to do anything like that? However, if the user is going to guide someone into using the system, things change. The user is in control now, he is the guide. He is the responsible for getting the task done.
You can get some experience from Let's Play channels on YouTube. Those people rarely talk when they are playing alone at home. But, when they have an audience - even a silent one - they become really talkative, explaining every little bit of what they are doing, mixing it up with some lateral commentary here and there. Their real audience, however, is just a camera, a lifeless electronic device nearby. Have you ever had a friend by your home while playing videogames, or even a little sibling? The effect is pretty much the same.
Don't ask them to explain - ask them to teach. To guide you. To show you stuff. Let them be responsible for the trip. The extra confidence will give you very nice results.