I think the heat is transferred from the "computer" to the LED itself. So the LED itself heats up.
To go into a bit more detail:
All physical objects emit thermal radiation (a.k.a. black-body radiation [edit, not quite correct]). The hotter the object is, the more radiation it emits and the shorter the wavelength of the radiation. For "red hot" objects, the radiation is in the visible part of the spectrum; for room-temperature objects, it's in the infrared, so you can't see it, but it's still being emitted. If you put two objects of different temperatures next to each other, they'll both emit radiation and absorb each others' emitted radiation. The hotter one will emit more radiation than the cooler one, so there will be a net transfer of energy from the hotter one to the cooler one (normally).
But in this experiment, when the researchers run the LED in reverse, this somehow suppresses the LED's normal thermal radiation emissions. The "computer" continues to emit thermal radiation, which is absorbed by the LED just like normal. But the LED doesn't emit thermal radiation back to the computer (at least, not as much as it normally would). So even though the LED is hotter than the computer, the LED acts like it's colder for thermal radiation purposes. So there's a net transfer of energy from the cooler "computer" to the hotter LED.
Of course, this isn't a complete solution; you still have to do something about the heat once you've transferred it to the LED. The article doesn't address this part of the problem. But I speculate you could just attach a heat-sink to the LED. Heat sinks work better at higher temperatures, so the computer+LED+heat-sink solution would be more effective than a computer+heat-sink alone.