Constant, invisible steering
Why Does a Moving Bike Stay Up?
A bike stays up because tipping makes the front wheel steer into the fall, curving it upright.
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Step 1 of 5
A still bike falls; a moving one does not.
Nothing about a stationary bicycle keeps it upright for even a second, yet the same machine rolling gently is remarkably hard to knock over.
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Step 2 of 5
Falling is corrected by steering.
Turning the front wheel towards the side you are tipping makes the bike curve that way, and the curve lifts you back over the wheels.
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Step 3 of 5
Most of the steering happens without you.
The forks are angled so the wheel's contact point trails behind the steering axis. Tip the bike and the wheel turns into the fall by itself.
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Step 4 of 5
The spinning wheels help a little.
A turning wheel does resist being tilted, but the effect on a bicycle is modest. Take the steering away and gyroscopes alone will not save you.
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Step 5 of 5
That is why learning feels sudden.
You are not learning to hold still, you are learning to steer into every wobble. Once that becomes automatic, balancing stops feeling like work.
Riderless bicycles have been filmed staying upright on their own above roughly 14 km/h, purely from their steering geometry.
The short version
The forks are angled so the wheel's contact point trails behind the steering axis, which makes the wheel turn into a fall by itself. The spinning wheels help only a little, and learning to ride is learning to steer into every wobble automatically.
Try it yourself
Roll a bike gently alongside you and let go. It stays up for a while, steering itself into each wobble, until it slows too much.