A motorcycle above walking pace turns by leaning, and the lean is initiated by momentarily steering in the opposite direction. Riders do this instinctively long before anyone explains it to them.
A two-wheeler turns by leaning, not by steering
In a corner the machine must lean inwards so that the combined effect of gravity and cornering force passes through the contact patches. Without lean, the bike would be pushed upright and outwards.
The steering angle needed to hold a given corner at speed is very small. Most of the geometry of the turn comes from how far the machine is leaning over.
So the real question is not how the rider points the front wheel, but how the rider gets the machine to fall into the lean in the first place.
Pressing forward on the inside bar starts the fall
Pushing the right handlebar forward briefly steers the front wheel to the left. The contact patch moves out from under the machine's centre of mass to the left.
With the wheels no longer directly beneath it, the machine begins to fall to the right, which is exactly the lean the rider wants for a right-hand corner.
The rider then releases the pressure and the front wheel settles at the small angle needed to hold the line. The whole sequence takes a fraction of a second.
Gyroscopic effects contribute but are not the whole story
A spinning wheel resists being tilted, and forcing it to steer produces a rolling moment. This effect is real and helps initiate the lean.
Experiments with counter-rotating wheels that cancel gyroscopic forces show that a bicycle still steers and balances, so the effect is a contributor rather than the mechanism.
The dominant factor is simply geometry: moving the contact patches sideways relative to the centre of mass is what makes the machine fall.
Why riders do it without knowing
Learning to ride is learning to balance, and balance corrections are made through the handlebars from the very first attempt. Countersteering is embedded in those corrections.
Because the input is brief and small, and because the rider is thinking about the corner rather than the bars, it does not register as a deliberate action.
Riders often discover it only when told, and then confirm it on the next ride by pressing a bar consciously and watching the machine drop into the turn.
Where it becomes a conscious skill
In an emergency swerve, a firm deliberate press produces a far faster change of direction than an instinctive attempt to steer towards the gap.
On fast sweeping roads, a rider who understands the input can set the lean angle in one movement rather than drifting into it over several corrections.
Riders returning after a break often benefit most from thinking about it explicitly, since the instinct fades more slowly than the confidence to use it does.