To set up the circumstance when you can do this is tricky. You usually have to put up with some degree of them repeating their conviction again and again before they understand what you are asking them for. There is always some disbelief that you are indeed interested and questions about what your motive is for wanting to hear their history. For them, there is some degree of risk in sharing a tender core belief story, stories which sometimes make no sense but are a largely emotional conclusion from sometimes a mysterious set of circumstances.
Also I'm sure that the risk is real; most people in this situation have had others plead for their story. If they dared to tell it or hint at parts of it, those more skilled debate advocates merely tore into the validity of what they had experienced. They were left with disillusionment without getting anything from the exchange to address their needs that the conviction provided. Also, to the extent they had to defend themselves from self-doubt, (which doesn't feel very good,) they may have invested a great deal of energy into "shoring up" their position - and may need to do more if they reveal their precarious ways of assigning meaning.
So because of this, some people will not, under any circumstances, risk telling you the truth and you cannot blame them. Many people cannot risk the truth from a various list of: too paranoid, too wounded, too unfamiliar with the ideals of dialog, not articulate enough to describe their own experience in hindsight or merely stupified that anyone would have asked.
The solution for these situations is to find many ways to reassure the person until they finally believe that you are going to, at least, not attack them if they tell their story - and, at best, once they can trust that you are in concert with them, maybe they will dare to accept some new exploration or interpretation of their story to revise their position once the two of you mutually define some new, more consciously chosen criteria to address these "essential needs."
However, some people do not know how they came by their conviction, so they have no "story" or definite experience to relate that they remember. The concept that people have an experience that leads to a conviction isn't something that has ever happened to them consciously yet. Or perhaps people may need to go back privately and think about their core belief experiences in a new way before they have anything to say about it as a story.
However, all this takes energy and time. For most people, it's better to just stay away from the topic entirely and not bother to open the "can of worms." Usually, most people just don't have the skills to deal with the history any better than the person who with which they sympathize with who is directly "under fire." So that's why people do not want to know other's problems.
The other reason, is sometimes when you do speak up, the upset people turn their attack from each other into ganging up on you! At least, this is what other people say is why they do not speak up in a group situation.