I was brought up with the religion so I guess there was never 'that' moment.
I can tell my mothers story though:
My grandmother was a devoted catholic and she had gotten a divorce because of my grandfather's infidelity. According to the Catholic church at that time, divorce was a mortal sin and so she would burn in hell for eternity. My mother could not understand how her mother, who had given everything to raise her children and who was such a gentle soul would end up in hell. (Later on the Catholic church would popularize special dispensation and so my grandma would no longer be going to hell by necessity.)
One day my eldest brother was playing in the driveway. At that time he was still a toddler and I was still a distant future accident waiting to happen. While looking out the kitchen window my mother noticed he was coloring something in with a crayon. My mother went outside and saw that it was a book. She thought it looked pretty expensive and so she took the book away (probably to his woeful disgust). Turns out it was Witness literature. She started to read and lo and behold she stumbled upon a section on the fate and state of the dead. She was instantly sold on the religion because now there was a way to solve my grandmother's eternal damnation dilemma.
Since that fateful incident the 'Truth' has dominated our family history to this very day.