The main reason this doesn't fly is because there is no expenditure of energy. You have energy stored in the magnetic field (and the stresses in the black bar) but stored energy does no work, and there are two problems that prevent that energy from being released in a way that causes flight.
The first problem is that the lower magnet can't move toward the upper magnet because there is a rigid bar holding it down. You might argue that with a strong enough pair of magnets, the magnetic attraction would overpower the normal force of the bar, but remember that the attraction applies to both magnets: if the lower magnet is attracted to the upper with a large enough force to lift the weight, then the upper magnet is being pulled down by the same force, and that force is transferred through the bar and pushes down on the lower magnet, preventing it from lifting off. The forces balance out and gravity takes over.
Let's say you do away with the bar though; now there's nothing holding the lower magnet down, and if the magnetic attraction is strong enough, it could lift off the ground. However, the new problem is that there is now also nothing holding the upper magnet up, so it will snap down towards the lower magnet quite quickly. Once the two magnets smack together, the potential difference (i.e. stored energy) is gone, so there's no upwards force and again gravity takes over and pulls the whole thing down.
So let's get around this another way: you tie the upper magnet to the bottom of a helicopter so that it doesn't fall down. Now the lower magnet is pulled up by the attraction, and the upper magnet is not pulled down, so the system is flying! However, it's not really the system that is flying, so much as the helicopter flying (and carrying the magnets as passengers). The energy stored in the magnetic field was released to bring the magnets closer together, but it takes the extra expenditure of energy from the fuel in the helicopter to continue to combat gravity and keep the thing aloft.