In case anyone is still lost think of it like this, first imagine a car.
A car moves because the wheels spin. They spin around, push against the ground and the car moves forward. If you put a car on a treadmill going the same speed in the opposite direction the car will not move forward. The treadmill pushes one way the wheels push another and the car stands still. That should be fairly apparent and intuitive.
Now let's add 1 thing to that car, lets make it like the batmobile. The force that now moves the car isn't the wheels. We have strapped a rocket to the back. The rocket propels air backwards, which pushes the car forwards. Even if the wheels were not moving at all the car would still move, assuming the rocket could overcome the friction of the wheels grinding on the ground. Now when we put the rocket powered car on a treadmill the wheels forces are still cancelled out. The wheels wont' push the car forward, but the force of the air being expelled by the rocket WILL move the car forward. This is how a plane moves. The wheels arent' what physically drives a plane down the runway, it's the force of the fans displacing air. So despite the fact that the treadmill cancels out the wheels propulsion, the plane will stil go forward. The treadmill speed is irrelavent, unless the thing can spin so fast that the the frictional forces on the wheels can counter the tremendous force of the air being displaced by the engines.
As long as the plane can move forward there is air under the wings moving, hence we get lift....
Overall what you need to realize is that the wheels spinning in a byproduct of the moving plane, not the reason the plane moves, hence the treadmill can counter the wheels all it wants, the plane will still go forward.