To feel the effect of countersteering, get on a straight section of road. At 30 mph or so, and without consciously doing anything else, lean to the left and then back to the right several times in succession This is how most novice riders steer. Try to get a feel for how long it takes for the bike to make these linked turns.
Next, at the same speed and without doing anything else, give a push on the left handlebar and then the right bar several times in succession. You should feel the bike react much faster. This is countersteering, and the harder you push the faster it will turn (up to a point). Take it easy, and compare it to just leaning alone. If you are doing it correctly you will notice you can turn much quicker.
Now experiment at higher speeds and compare the two methods. You are now on your way to a more efficient method of turning a motorcycle. Continue to practice countersteering until it becomes ingrained and you don't have to think about it anymore.
Congratulations, you are now a safer rider because you can swerve away from danger quicker than you could before.