Basically, they applied current to the plastic sheet and it created pockets of air underneath that moved from the front to the back, moving the sheet somewhat like a stingray in water.
They use the term 'flying' in quotes but it moves about a centimeter per second and this is not a research article so I will go ahead and say this thing flies.
What would make it more awesome? They're working on a solar-powered version.
Citation: Noah T. Jafferis, Howard A. Stone, and James C. Sturm, 'Traveling wave-induced aerodynamic propulsive forces using piezoelectrically deformed substrates', Appl. Phys. Lett. 99, 114102 (2011); doi:10.1063/1.3637635