A few years ago when I was learning how to do customer research, my manager at the time said it's just as important to know when to be quiet as it is to know which questions to ask.

That applies to interviewing someone, for sure, and it applies to any time you're talking to someone with the purpose of learning from them. 

You get more and better information if you know when to leave pauses in the conversation. Pauses let the person collect their thoughts and decide if there is more to say on the topic. Really, the person asking questions should be talking less than the person answering questions. As the interviewer, you want to guide the conversation. Not dominate it. 

Getting used to those pauses might take some time. I tend to be comfortable with silent moments, but I know lots of people aren't. With practice, you learn to navigate the pauses.

One more note: With doing interviews for customer research and marketing, I don't like using the word "interview" even though that's technically the right word. "Interview" sounds too formal, like I'm a journalist and going to quote the person word for word. I'd rather set a more casual tone. That's more comfortable for me and for the person I'm talking to. Instead of "Can I interview you about...?", I say "I'd like to learn about [topic]...from you." That sets up an easy conversation.