In the UK this is called "counter steering" and a lot of people can't get their head round the idea of steering left to go right
I don't exactly know how it works, but, my theory for what it's worth is that when you push, say on the right handle bar the front wheel is pushed to the left, effectively acting as a rudder, and steering the bike to the right. It's the only explanation I can think of.
I also think everybody counter steers, even if they don't know they're doing it. The very act of leaning the bike over involves pushing the handlebars down towards the side of the road the same side as the bend you're going round, if you follow?? think about this, do it now in your chair, your near side to the bend hand will go forward.
How much do you push? when you've learned to consciously counter steer and are confident with it, it can be a life-saver. If you've committed to a bend and find yourself "running out of road" push harder, you'll get round if road conditions allow. Remember, it doesn't matter how capable you are, or think you are, the bike is always capable of more, unless your name is Rossi
Watch those chicken strips vanish