Temperature has a lot of different definitions. It can be the average kinetic energy of the particles of a system, or the change in heat with respect to the change in entropy (This can be derived through both thermodynamic and statistical analysis).

In the paper, they arrive at their conclusion of "negative temperature" by manipulating the distribution of energy, such that the particles are more likely to occur at higher energies. Normally, this distribution would only be transient, and the crystal lattice of particles they had set up would collapse, eventually giving the distribution on the left side of that image. However, through some strange quantum magic that I do not understand, they managed to keep that energy distribution stable.

In the last paragraph of the publication, they give a short comment that the relevance of these temperature definitions is still being investigated:

In ultracold quantum gases, the basic concepts of thermodynamics, positive or negative temperature, or whether a temperature concept is even relevant, are under intense and profound exploration.

If you want to read the actual publication and need past the paywall, PM me.