I hope we can all agree that a scooter, or any two wheeled vehicle for that matter, is turned, at speeds greater than approximately 15 mph, by leaning. Under 15 mph it can be turned by steering the front wheel in the direction you intend to go. Agreed?
So, in the simplest of terms, counter-steering is a method to force your two wheeled transportation to lean, thus initiating a turn.
I will leave it at that for now...