Countersteering is purposefully turning to the left and NOT leaning to the left, causing you to begin to topple over to the right...but then you quickly save it by turning the bars to the right. Now you are turning and leaning to the right.
https://www.youtube.com/watch?v=ljywO-B_yew
Check it out.
The part I find amazing is you understand how to make this work in a part of your brain that apparently doesn’t communicate with the conscious understanding part. People can ride a motorcycle and not even understand how to make it turn, yet they do it.
Being able to do a thing is vastly different from being able to explain how to do a thing. If you ask me how I turn on a bike, the real answer is I just have the intention to go that way, and then I start going that way. Lots of other things are like that too; ask someone who is fast at word searches how they do it and it'll be something like "I kind of look at the grid and the word(s) pop out; and if it gets really tough I move my head around a bit and then they pop out" Not very helpful if that's not what happens to you, but it's what I do.
The simplest way to understand it is to consider that if the bike did not do this, it would be unrideable. The bike tends to return to center. If turning the wheel right initiated a right turn, the bike would just fall over.