The key statement is that the conveyor is moving in the opposite direction as the airplane, at the same speed as the airplane. So when you apply thrust, and start to accelerate the conveyor starts moving. Your tires start rotating, but since the conveyor speed isn't based on the tires, but the aircraft it actually won't start moving backwards until you've overcome friction and are moving the aircraft forward. The wheels will turn faster than normal, but the aircraft will still takeoff. The thing is, it'd have to be a huge conveyor because it's going to take the same length of runway for it to take off.
If the conveyor isn't motorized, and is free-wheeling and frictionless, and moves rearward in conjunction with the speed of the tires (not the aircraft), then you won't take off.
I'm thinking everyone is confusing the second case with the first.