Think of it this way: you're at the airport. You're running late, so you drag your carry-on wheeled bag down as many of the moving walkways as possible to save time. In this example, the bag represents the plane and you are the plane's engines, providing the thrust. The walkway is independently powered, doing its own thing whether you're on it or not.
As you pull the bag down the walkway, the wheels on your bag turn. They're rolling over the walkway, at a speed relative to the walkway surface the same as if on flat ground, but the speed of the bag's progress is increased in relation to the hallway outside the belt (assuming you're walking with the belt). So, for the sake of argument, lets say that the belt allows the bag to progress along at twice the speed it would on flat ground, with the wheels rolling at normal speed. So, the wheels are rolling at half the rate otherwise required for that rate of progress. Now, if you walk against the belt - thereby slowing your forward progress down and pissing off other airport visitors that are coming at you head-on - your bag's wheels are still rolling at the same rate in relation to the belt surface. But your pace of progress is cut in half. If you had enough "thrust" to get back up to your walking speed against the belt (relative to the outside world), your bag's wheels now spin at 2X the RPM as they would on flat ground for that speed. But you still make progress - you still achieve your goal of walking to the end of the belt. If you happen to have wings, and in the process of getting to the end of the belt happen to go fast enough to achieve sufficient lift, you take off.
Its a matter of relative motion and speed. And, with the assumption of an independent belt system (I read nothing to indicate that it is somehow driven by the plane itself), the wheels react to that. But that isn't part of the closed-loop that thrusts the plane forward and up to speed. If the belt is at a fixed speed, assuming the wheel hub friction is less than the belt drive mechanism, as the plane speeds up, the wheels' speeds are only offset somewhat from their normal rate of rotation at take-off. The wheels roll down the moving belt, just like the bag at the airport that's trying to get to the plane. Now, if the belt is jiggered so that it speeds up in proportion to the plane's acceleration, the wheels will continue to roll at a rate 2X (or whatever) their normal rate, increasing in speed as they do. But the plane is thrust along and, as a system, doesn't care what the wheels are doing. There's nothing in the plane-belt system that causes the plane to drive the belt as described in the problem.