To expand on the above post:
Take what Ext is saying and add this to it.
That wheel/tire combo acting like a gyroscope.....now, force it off it's axis (push on the handlebar), the bike will "fall over" off it's axis, and follow the small side of the tire. Much like if you were to take a paper cup, lay it on it's side, and give it a push. What does it do? It turns in a circle. Same principle with the motorbike.
It is a hard concept to wrap your mind around, but the simplest way to "get" it is think this: "Push right, turn right." "Push left. Turn left."
As far as the debate goes about the bicycle......The MSF instructor was right to a degree, and Rowdy was right to a degree. Apparently, that is an MSF teaching maxim, I don't know, I heard the same thing at my MSF class. I ride a bicycle every night, and not just tooling around either. Maybe the MSF believes that the average Joe won't pedal a bike fast enough on a consistent basis for countersteering to be relevant. However, you do need a speed far faster than just above walking speed. I have to be hauling the mail on the bicycle when I'm able to countersteer, but up until that point, if I push right to go right, I turn left.