How do you start that conversation? Because it seems rather difficult to get just some random person to tell you their problems at work or at home.
That said... start with empathy. Put yourself in your prospects shoes, so to speak. Think about the things you will talk about with someone you don't know, the things you won't, and why they differ. Further consider how you decide who you will and won't talk about certain things with.
Example: let's say you're a guy and you have ED. Most guys consider this incredibly embarrassing and wouldn't talk about it with many people. You might even be reluctant to tell your doctor. But you probably would, eventually. OTOH, you might be sitting at a bar, have a stranger sit down next to you who asks "how's it going", and then you blurt out "Ah, hell, my dick doesn't work, the transmission in my car is shot, and my dog died. How about you?"
So, why would you tell your doctor about your embarrassing condition? Probably because your doctor has massive credibility in your eyes and represents a source of a resolution to your problem.
But why tell the stranger in the bar? He/she can't help right? Maybe not, but since they are a total stranger that you'll probably never see again, you can safely tell them without caring one whit about their judgment of your condition.
So, maybe demonstrating credibility and a non-judgmental attitude are keys to getting people to talk about things? Granted, this is an over-simplified example, but think through some similar scenarios and see where it leads you.
Regarding credibility: a lot of this is tied up in the research you've already done (you did do preliminary research, right?) related to your prospect (in terms of industry vertical, demographics, geography, etc.), and the questions you ask. Asking good questions is a good way to demonstrate that you know what you're talking about and may represent - as your doctor does - a viable source to some mitigation of the problem at hand.
Beyond that: many/most books written on the topic of selling and sales cover this ground to varying degrees. Search the HN archives for the (many) threads on "sales book recommendations" and peruse some of the titles. There are also a lot of books written to target consultants or would-be consultants that deal with similar material.
There's an idea that pickup artists talk about, called "enter her world". The notion goes something like this: if you're out at a bar, and you're a guy, and you see an attractive woman you want to talk to, you approach her and start a conversation, at which point you're just "another sleazy guy in a bar". But if you quickly demonstrate, using your questions, statements, stories, etc., that you have some understanding of the woman's world, the issues she has to deal with, etc., then you can "enter her world" and now you're not just "a sleazy guy at a bar". Now the barriers go down a little bit and the conversation becomes more personal, and more meaningful. OK, it's not a perfect analogy, but you can probably see the connection. The point is basically that you have to earn admission into someone's world if you want them to talk about things beyond a superficial level.
I found it always help to explain your goal/project your working on. Starting the conversation with: “Hi I’m working on a fun project and wanted to get out and talk to some people about it. Do you have some time to help me?”
After doing this with 30 people in your social circle, slowly start reading up and improving your approach and start asking people further away from you
No relation to the author, I don't make money off the recommendation, etc.
Two books I would recommend are
How to Win Friends and Influence People
And
The Mom Test