When the wheel pitches to the either side, the road under the bicycle pushes the wheel back to center.
When the bicycle leans to the side, the wheel pitches as well. Now the road under the bicycle pushes the wheel back towards center, but the angle of the tire to the road is also skewed, so the some of the force also gets translated into pushing the bike upright.
It's a little reminiscent of the self-centering action you get when you have a double-cone (or cylinder tapered on both ends) rolling down two rails.
I think if you fixed the front tire so there was no steering, you wouldn't get any stability from speed.