Countersteering is the turning of the handlebars against the direction of the turn; for a leftward turn, the handlebars are turned to the right, and for a rightward turn they are turned to the left. Thus, the cyclist pushes on the handle on the inside of the turn and/or pulls on the handle on the outside. The goal of countersteering is making the bicycle lean to the inside of the turn. While doing this, it not necessary that the front wheel is visibly turned. What matters, is that pressure is put on the handlebars in the opposite direction as the direction of movement.
The importance of countersteering for leaning the bicycle increases with speed and weight. At low speed and with a light bicycle, body weight displacement is usually sufficient for taking the turn. At high speed and with a heavy bicycle, the bicycle is more stable and counter steering is necessary to make it lean. Countersteering is briefly explained in the first minute after the start of the video below (in Dutch). Contrary to the commentator’s claim in the video, you cannot see that Fabian Cancellara is countersteering, but he is doing it. And the scientist that explains countersteering only mentions the pushing on the inner handle and not the pulling at the outer handle.
The video below explains how countersteering works. At the end of the video, a link is made with the intrinsic stability of the bicycle, from which follows that some countersteering is also required during the turn.
If you don’t run away for explanation in physics terms, then you will certainly appreciate the video below. The demonstration is given using a motorcycle, but this does not matter for the principle; a motorcycle behaves like a heavy and fast e-bike.
Countersteering is especially important for cyclists that replace their regular bicycle by a faster and heavier e-bike. In fact, countersteering becomes more important as the bicycle becomes faster and heavier. Beginning e-bikers therefore must practice countersteering.