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Host John Dehlin (featured caricature) and others, seek to answer this question in the Mormon Stories Podcast. The show typically consists of John or another host interviewing various mormons and their experiences. The guests range from people who have left the church and have embraced different belief systems like atheism or even evangelical christianity to LDS scholars like Richard Bushman (author of Joseph Smith-Rough Stone Rolling) and Dr. Ted Lyon (retired BYU professor; currently President of Santiago Chile Temple). Some episodes range in content from homosexuality to uncomfortable church history to what to do when one spouse stops believing while the other remains a believer.
I don't want to give the impression that this podcast is negative though. John Dehlin is an active member of the church and in good standing. With the exception of those who have left the church, which seems to be the smallest group, most of his guests are also active members who at least find peace in the church and even believe in the church. What it does, however, is expose members to many of the less comfortable aspects of the church but in a way which does not seek to damage faith but create understanding in a [sometimes brutally] honest and very real way. It also attempts to build a bridge to those who have lost their faith through this understanding.
So, how do we help these "lost sheep"? There is a tendancy in the church to shun people who leave by choice. I think it's almost a tribal instinct. The natural man. Simply put, we love them. We accept them. We don't fear them. Respect their choice to leave if that is what they choose to do. Support them if they wish to stay or return to the fold. In short, don't judge them. Let them know that whatever they choose to do, you will still be there for them. This is the spirit of what John Dehlin and Mormon Stories is about.
So what are some of the issues people deal with which can lead someone to leave the church? This episode gives a pretty good overview if you're willing to go there, as well as a more complete section on how to help:
Why People Leave the LDS Church and How We Can Help