8) I was an MSF instructor for over 20 years. During that time I did a lot of reading about "countersteering" and just what the mechanical (physics, etc.) action was that caused it to happen. I also discussed it in detail with a few of the mechanical engineers with whom I work.
The front wheel, once it starts spinning with any real speed actually becomes a gyro. The rear wheel is also, but its not hooked to a steering head. When you try to turn a spinning gyro, the actual action takes place 90 degrees later. It's called gyroscopic precession. So, when you turn a spinning motorcycle wheel (gyro) around a vertical axis, say to the right (push left), what actually happens is the this: the force on the wheel will cause it to rotate around an axis 90 degrees past where you think it should. When it does that, the top of the wheel will actually move to the left as the turning axis for the axle is as if the forks were pointing straight out in front of the bike. I know it sounds strange, but these are the simple facts. There's also much more happening in the physics of the motorcycle frame, but that's enough to make you understand.
Did we ever explain this in the class? Hell no! Most people look at you like you've got three heads when you start to talk about this.
Now as to this being the only way to turn a motorcycle. Not really. One way was mentioned above in using weight transfer. It will work but it's harder to accomplish on many machines, and is slower to react. You can also use your right knee with pressure on the gas tank to turn left, or the left knee to go right. Again, on most machines it really not good for quick reaction. The high performance sport bikes will respond to the knees and the weight transfer on the pegs fairly quick due the quicker steering geometry built into them. But most other bikes, and motor scooters, aren't as quick.
You really don't have to believe any of this to get it to work. As we preached, excuse me, as we intensely instructed, Look left, Push left, Go left; and the opposite for the right. Believe it, use it, and enjoy it. It's the easiest, quickest way to get a two-wheeled machine to change direction.
And, if you're so inclined you can do the research to better understand it.
Good Luck!