Leave it to @CaptainObvious to post something that's not so obvious.

I must disagree.

First, it's precession, not procession, paired with the "caster effect" and static friction. It's a self correcting system that only works when in forward motion.
This nonsense with counter rotating wheels countering the gyroscopic effect ignores the fact that only the forward rotating wheels are using friction to direct the path of the bicycle based on their angles. It's not JUST the gyroscopic effect, that only determines the resistance to angular change, it's friction directed by gyroscopic precession and the self correcting caster effect. Come on.

Nobody knows how we ride bikes?! (A different claim from we don't know why they can remain upright when ghost ridden) Nonsense, it's balance plain and simple. If you don't keep the center of gravity exactly above the contact points with the ground, you tilt. When the centrifugal force exceeds static friction of the tires you slide out or when gravity exceeds the opposing centrifugal force caused by turning, you fall. Try turning your bike but staying exactly upright, no lean. Now, when you heal, try leaning without turning, lock the wheel straight, you'll fall again. It's multiple forces in concert.

I think a decent physics teacher would wipe the floor with this. It ignores so much to make these "we just don't know" claims.

*wheels

Adding video to channels (Wheels) - requested by ant.