I was fortunate enough to land on a video, thanks to My Shaved Legs and user IanJSeattle.. As unconscious a behavior is walking and breathing to humans, so is counter steering to the cyclist (oh of course, most cyclists are human too unless you beg to differ). I find it slightly ridiculous that a big deal is made out of this principle in most cycling books, when in fact each one of us uses it pretty much on a daily basis, whenever we ride.
To handle a corner, you have to effectively lean into a turn so as to shift the combined center of mass of you and your bicycle. If you don't lean, centrifugal forces will disturb the balance of forces and throw you to the ground.
How counter steering works : Source
But the only way to cause that lean is to move the support points in the opposite direction first - a tiny jolt of the handlebars for a fraction of second depending on speed that we aren't really conscious of but it exists, as this video proves with a slow motion camera. The counter steering phenomenon becomes evident when there is an obstacle preventing the wheel from counter steering, such as an overlapped wheel which is a common cause for crashes in bicycle races. Next time, save your bones and equipment by being aware of the need to counter steer.