By slightly turning the handlebars right or left, you impart some of the rotation of the front wheel (“angular momentum”) to rotate the bike around its long axis, the direction in which it rolls.
That’s it, the whole story.
While it is true that the precession of the spinning front wheel in response to an input steer torque is about the roll axis, this effect is minuscule in comparison to the roll moments generated first by the laterally accelerating contact patches and then by gravity acting on the center of mass that is no longer directly above those contact patches. Dr. Stern’s explanation also implies that gyroscopic effect is necessary for balancing a single track vehicle by steering it, which will come as quite a shock to the guy riding this thing:
As for the actual magnitude of the contribution to the overall roll moment on a bike from the precession of the front wheel, we can refer to the nice example provided by Professor Cossalter in his excellent Motorcycle Dynamics.
On page 304 of the second edition, he calculates the roll moment generated by gyroscopic effect for a motorcycle traveling at 22 m/s (79 km/h or 49 mph) to be 3.5 N-m (2.6 lb-ft) and compares it to the roll moment generated by the accelerating contact patches of 30 N-m (22 lb-ft), which is 8.6 times larger. He concludes with the note that the gyroscopic effect is present from the instant torque is applied at the handlebars, and the roll moment generated by the lateral force of the tires can take some time, ~0.1 seconds in this example, to build up. We should expect the gyroscopic effect on bicycles, with lighter wheels and lower speeds, to be proportionately smaller.
There is a little more nonsense at the end of Dr. Stern’s reply about how bikes with straight forks, such as motorcycles, have no fork offset and so more trail, more stability, and less agility.
The design of the front wheel “fork” is quite interesting, turning forward like the letter j. This is an “inverse caster” which makes the ride less stable–but allows a skilled rider nimble moves. If you ever look at a motorcycle, or at the bikes used in a circus on the high wire, they lack this “letter j” feature, the wheel axis is always in line with the handlebar shaft. That helps stability but reduces agility.
Can you imagine how much better these guys would be if there were some way to reduce trail and increase agility without curved forks.
Of course curved forks is only one of several ways to generate fork offset and thereby alter trail.
No, bicycle science is not rocket science, but it still requires that you do your homework.