In this case the plane isn't much different than a glider. A glider gets towed either by a powered aircraft or a winch/catapult type mechanism on the ground, and when the airflow over the wings is fast enough to create adequate lift, the glider flies. Or maybe better to compare to a plane getting catapulted off an aircraft carrier, where the catapult accelerates the plane to a speed where the wings develops lift, and then the thrust of the engines takes over to keep the plane flying at a speed required to maintain the lift.
If you put an airplane with the engine not running on a platform and accelerated the platform fast enough, the plane would start to fly after developing lift, and the plane would glide back down to the ground, or stall and crash into the ground depending on circumstances and how fast the pilot could react on the controls etc.
I thought the old plane on a treadmill theory had got beaten to death years ago, even Mythbusters did an episode on that.