I must have caught the bug from my mom. She had an old Schwinn Breeze with a shopping basket on the front and a kiddie carrier on the back. Also, we were a 1-car family. So I remember riding on various errands around town. To this day, the "tick tick tick" of a Sturmey Archer hub still brings back memories, and I have those hubs on a couple of my bikes.
I think I liked the mechanical aspect of it, plus just the freedom of being able to go out on my own and do things. Driving the family car meant haggling with everybody else who needed to use it. I didn't get a car until I was halfway through college.
I'm still not a competitive rider, but enjoy using a bike to get around town and go on moderate weekend rides. And I still love tinkering with old bikes.