I thought I had a decent understanding of physics. But after reading this, someone should revoke my 'amateur scientist' card. This (from Wikipedia) helped me understand it:
Since we started with over half the atoms in the spin-down state, initially this drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point more than half of the spins are in the spin-up position. In this case, adding additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature.
This is not about the common notions of hot and cold, this is thermodynamic temperature, which is about entropy and energy. OK, so to me, this amounts to a neat physics trick. The universe will not collapse and our understanding of physics hasn't changed. Hopefully, it'll inspire a few folks (like me) to understand it better. Doing more reading, it reads like this only works in a 'system' of atoms, not with a singular atom. If you can't make a single atom colder than 0.0 kelvin, then the fundamental physics models don't change. Somewhat frustratingly, I see no mentions of anti-gravity, which was the most radical thing said on the nature.com article. So, please erase that and any anti-gravity belt predictions from your memory banks.
Speaking of theoretical maximums, another fascinating one is the concept of "absolute hot", which is currently defined as the Planck temperature, (1.416785×10^32^ kelvin). All physics models break down, even things like gravity. And, theory says that the entire universe has already experienced this temperature, a fraction of a second after the big bang. Enjoy that thought.