My sense is that people value the truth to varying degrees. Further, people encounter barriers to pursuing the truth to varying degrees. Whether or not someone ends up here is likely a function of them caring about truth enough to make the relevant social and psychological sacrifices to get past the barriers.
For me, I don't remember when I started caring about whether or not my beliefs were true. I know that the moment the possibility of God's non-existence was put to me I immediately became an agnostic- and an atheist when I learned about the scientific method, Karl Popper, etc. I was raised Catholic ostensibly but my mother is a Unitarian (though one who believes in a fair bit of New-Agey gobbledygook) and my Catholic father is a doubter and extreme skeptic. The areas I've lived in have always been fairly non-religious and relatively non-Christian (until I attended a Catholic university).
The answer to the question "what got the transition started?" is probably a just knowledge of the rationalist position and hearing an unbiased version of rationalist arguments. What made the transition possible was valuing truth and having few significant barriers to pursuing the truth. What makes people value truth, I suspect, usually comes before most people's conscious memory and not recognizable at the time.
However, I did have an experience that increased the how much I valued truth-My parents got divorced and told me contradicting stories. Hypothesis 1: Being lied to increases one's subjective value of truth. Hypothesis 2: Being lied to by people who answered all of your initial questions and guided your initial decisions increases one's subjective value of truth.
To break up the awkward silence at the start of a recent Overcoming Bias meetup, I asked everyone present to tell their rationalist origin story - a key event or fact that played a role in their first beginning to aspire to rationality. This worked surprisingly well (and I would recommend it for future meetups).
I think I've already told enough of my own origin story on Overcoming Bias: how I was digging in my parents' yard as a kid and found a tarnished silver amulet inscribed with Bayes's Theorem, and how I wore it to bed that night and dreamed of a woman in white, holding an ancient leather-bound book called Judgment Under Uncertainty: Heuristics and Biases (eds. D. Kahneman, P. Slovic, and A. Tversky, 1982)... but there's no need to go into that again.
So, seriously... how did you originally go down that road?
Added: For some odd reason, many of the commenters here seem to have had a single experience in common - namely, at some point, encountering Overcoming Bias... But I'm especially interested in what it takes to get the transition started - crossing the first divide. This would be very valuable knowledge if it can be generalized. If that did happen at OB, please try to specify what was the crucial "Aha!" insight (down to the specific post if possible).