The front wheel is on an angle called a castor.
Leaning the bike to one side causes the tire to steer in that direction due to the offset steering angle. Try it by holding the bike still and upright, just let the bike lean over a bit and watch the tire turn on its own.
If falling to the left, the bike also steers left. And when you’re going forward, intertia will push the weight of the bike and rider against that lean.
You’ve felt that intertia before. Most commonly when riding in a car. The car takes a turn to one direction and you feel pushed towards the opposite direction.
Since that push of intertia is against the direction the bike leans, the intertia is constantly working to correct the lean and push the bike back upwards, typically.
But if you’re not moving forwards fast enough, there’s not enough intertia to push you back upwards. So you just fall over.
There’s a little more to it than that. A person riding a bike can influence the steering, so they can contol the way and direction the bike turns and leans, often might be called counter steering. A bike doesn’t actually require that castor angle to stay upright if the rider just does the corrections manually. But that intertia from moving forwards (or backwards) is generally required.
You can also picture trying to balance a broomstick on your hand. If the stick starts falling to the side, you move your hand and the bottom of the stick over to underneath the stick’s center and “catch” the stick before it falls over.
If the bike starts falling to the side, you steer to move the tires underneath the bike’s center and similarly “catch” it before falling over.
When turning, you allow the bike to lean over enough that it doesn’t fully fall over. But you also don’t fully “catch” it. You let the bike keep falling so you can continue turing and moving the wheels underneath it as you go through the turn.
It’s not until you’ve completed a turn that you would fully “catch” the bike and get it back upright again.
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