How do you get customers to share more than they normally would when you ask a question during discovery?

The first action is to set and agree the purpose of the meeting
"Ms customer the purpose of the meeting is to:
Briefly explain the progarms and services we offer
Ascertain your needs
See where there's a match and agree the next step"
The ask if they wish to commence with what the seller does or the customer's needs
Once they agree to the purpose of the meeting then there is a tacit understanding that the seller is expected to ask questions. 
There may be many reasons why the customer is relunctant to share information, including the timing be inappropriate and/or it is the wrong question.
Salepeople can ask a 'closed' question (Answer will be a 'yes' or 'no' or a fact) when they wanted an 'open' response (Wanted to elicit thoughts, desires, feelings, etc)
Example Seller "Do you think that it would be a good idea to explore this possibility" Customer "Yes"
Another possibility is that the customer feels that the questions are  leading to an outcome the seller wants so they feel pressured. 

Ask the right question, make it about them and their business needs. “When considering a solution that would solve for your ____. What would be the important characteristics that would really make a difference to your team?”  This gives a glimpse of decision making criteria. Don’t forget to say “anything else?” At the end of their list. 

Adding to what the experts below suggested, here are some generic questions to build a line of thought. "How are you doing this today" "How can this be better?" And as the others said...then be silent, let them talk. Your job is not to talk but to listen, drive the conversations towards the next step with just asking questions and listening. Listening means to understand the pain behind what the other side says (or when they ask, "the question behind a question").  Also important is how you ask, this can be learned. There are methodologies and techniques (Active Listening, Solution Selling, Black Swan, SPIN and more) one can learn so the prospect feels comfortable during the whole conversation and easily wants to move to next steps. 

Practice getting comfortable with silence. In teaching we sometimes use a '3-second' rule after asking a class a question. Counting to 3 can help prevent trying to fill the silence too soon. For "more than normal sharing", customers often need some time to think through and decide if they want to share something 'more than normal'.

also agree with the great points Steve Peterson makes on trust and searching for pain. 

It's easier said than done but starts with building trust and searching for pain. Understand your customer at a deep level prior to the call. The challenges they are facing, the desired business outcomes. Don't ask questions you can find the answers to with a bit of work. Lastly, ask open ended questions that tie back to the challenges or desired outcomes. If you can find a pain point, that’s when the conversation starts.