First of all, nantennas in general don't violate the second law of thermodynamics, so they are not perpetual motion machines of second kind. As long as the total entropy goes up, the second law is obeyed. In other variables, it really means that a part of the incoming heat has to heat the nantenna up but there may still be a lot of energy left for energy production, much like in any other heat engine.
The Wikipedia suggestion that natennas could violate the second law only referred to a particular application hypothesized by Mr Novack. If he could be cooling the room while getting energy out of it, and if the gadget to cool the room were not connected to any cooler heat bath, then it would indeed be a perpetual motion machine of second kind and it would be impossible.
The reason why Nature makes it impossible is kind of trivial. If the room has temperature $T$, then the nantenna or "power plant" may only be kept at the same temperature $T$ if there's equilibrium. But if that's the case, the nantenna emits thermal radiation, too. So even if it absorbs some incoming radiation, it still radiates its own. They're balanced and the energy gain is zero. Solar cells and "legitimate applications" of nantennas can only create energy because they work with incoming light whose "own" temperature is higher than the temperature of the solar cell or nantenna itself. For example, solar radiation has the temperature comparable to 5,500 Celsius degrees.
The solar cells are effectively heat engines operating between this high temperature and a much lower temperature of the ground. The same is really true about life on Earth, too. The energy from the Sun may be converted and is often converted to useful energy or work because the high-energy photons from the Sun – which correspond to a high temperature and therefore a low entropy per unit energy ($E\sim TS$) – are processed on Earth and the energy is finally emitted in much lower-temperature "infrared" thermal photons – which carry a higher entropy. So the entropy can go up even if a part of the incoming energy is converted to useful work. The temperature inequality between the solar surface (and the solar radiation) on one hand and the cool temperature of the outer space is necessary for the Sun to play this often praised beneficial role.