How can an LED have more than 100 percent efficiency
This is a great example of how science reporting can be sensationalized.

This isn't at all near the 230% efficiency that the Gizmodo article headlines.

In semiconductors there are three primary energy transport systems: Electrons, Photons and Phonons.  Phonons are simply how heat (i.e. lattice vibrations) manifests itself within a semiconductor.  Heat is vibrations and within a regular lattice these vibrations end up being quantized.  However, they behave very strangely with all sorts of interactions (light does not interact with itself - phonons do).  And phonons and electrons and photons all interact in a this quantum soup.

So what is happening here is that there is some energy being "borrowed" from the phonons in the lattice.  In fact this is well known and the Phonon/Photon interactions are well understood and modelled.

The original paper is just providing experimental confirmation on an effect that has long been anticipated.

* The analogy
Houses now a days often have heat pumps, that pump heat from the house into the ground in summer and then pump heat from the ground to the house in winter.  These are often rated at higher than 100% efficiencies too.  What that rating is saying is: "if you supply X amount of power to the unit to pump the fluids around- we will extract 1.5X the heat into or out of your house".

So it is not wrong to look at this similarly, think of the electrical energy as being the energy used to move electrons across the bandgap that then allows for phonon interaction to add energy to the emitted light in the form of additional photons.