I have noticed that The Moth style events (story slams etc.) are pretty much entirely personal stories and primarily a younger crowd than a typical storytelling event. I think this stems from what others have said about how people feel disconnected, don't have the same opportunity to tell their own and hear other's stories as in the past. We are also all coming from much more diverse backgrounds with different experiences than ever before. I think in this regard that personal stories serve two purposes. Hearing the stories of someone from a similar time, place or culture as you helps to reinforce your personal identity. Hearing the stories of someone from a different time, place or culture is new, exotic and hopefully helps you to begin identifying with them.
Perhaps the appeal of The Moth's aesthetic (ten minute true stories from the teller's personal life (although recent squabbling in the blogging community over the veracity of Malcolm Gladwell's "true" story at The Moth seems to indicate that they don't let the "true" part get in the way of a good story) is this: the simple elevation of the well-done kitchen table story with a microphone and a spotlight (in a venue with a well-stocked bar) celebrates America's obsession with the cult of the individual.
And I suspect the resonance for The Moth's audiences is not just similar "time, place or culture" (because they go out of their way to find tellers with odd jobs and unique experiences to share) but similar reactions to experience. That is, it's the emotional content that resonates, not necessarily the contextual details. And in doing this, it validates the idea of the individual. (Although, the concept of "Everybody has a story to tell" is by no means exclusive to nightclubs in the Village)
And again, if you've got an underused storytelling muscle in your frontal lobe, it's easier to visualize a story set in a familiar milieu, be it New York in the 1990s or Middle American in the 1950s.