The original response to this question was just terrible.
Because modern bicycles are equipped with a pair of gyroscopic stabilization devices that require the motion of the bike in order to operate. These devices are known as “wheels.”
It implies that the reader is a dummy for not knowing this obvious “fact” and that non-modern bicycles maybe didn’t have a pair of wheels.
So why do bikes stay up? The answer is: trail.
While it is true that both trail and gyroscopic effects can contribute to the self-stability of a bike, it has been demonstrated, by Kooijman, et al., in their 2011 Science article, that neither trait is necessary nor sufficient by themselves for self-stability.
Instead, the simple answer is that when a bike is moving forward, it can be steered to keep the wheels under the center of mass. More correctly, by steering in the direction of a lean, the tire contact patches accelerate in the direction of the lean causing a moment about the center of mass to counter the moment created by gravity.
The steering torque necessary to steer in the direction of the lean can be provided by the rider, or, in some cases, by some combination of mass distribution, geometry, gyroscopic effects, and tire properties.
The revised response ends with
On the bike path of progress, you have to expect the occasional bum steer.
They have provided two bum steers so far. Perhaps they can get it right on the third try.