Without getting all quantum about it, it seems to work by configuring the graphene as one electrode (or both) of a capacitor. As the graphene flaps in the breeze (actually Brownian motion, I assume), the changing geometry will change the capacitance. If there is a charge on the cap, then the voltage would flap too, inversely (Q = CV). A changing voltage would tend to push about a current (i.e. charge motion). If their diodes somehow have a negligible forward voltage drop, then brief moments of higher voltage on the flappy cap would push increments of charge onto the storage cap (the one to the right of the diodes) until the storage cap voltage rose to equal the peak flappy voltage (minus the diode voltage drop). When the storage capacitance is greater than the flappy graphene capacitance, the stored energy will be greater than the tiny increments (energy = 1/2 CV^2). Then every so often the storage cap could be switched to drive some load.

So why would there be a charge in the flappy capacitor? Well, a bias voltage is applied by this battery in the circuit. Ahem, before invoking any principles of thermodynamics, I'd like to see some detailed measurements, or even careful theory, of the current out of the battery and back into the battery, particularly w.r.t. the instantaneous voltages (P = VI), to prove that the time-average power delivered by the battery is precisely zero, or at least far lower than that putatively transferred to the load (light bulb in the animation). Otherwise it's behold, we found a high-resistance path for our battery to run a light bulb.

There is some precedent for semiconductor thermal magic in the Peltier junction. It makes one end colder and the other end hotter by applying electric watts to it, with no moving parts aside from the electrons and holes recombining, or something. But nobody sees thermodynamics being violated or extended there.

But here the magic is to take away thermal energy from one place, the flappy graphene (thereby cooling it) and moving the energy to the light bulb (thereby heating it). It seems you could then use any old heat engine to extract work by letting the heat flow from the resulting hotter place back to the colder place, getting "limitless power" for free from your perpetual motion machine. Conceivably it's a sort of heat pump that gets some multiplier above the battery power. But I think the burden of proof is on the Arkansas folks to explain how this can be. If I had more energy (no pun intended, really) right now I'd try to find the paper that the press release is based on, to see if there's a little more truth there.

Meanwhile, I doubt it.