Negative thermodynamic temperatures are "hotter" in the sense that energy will spontaneously flow from a system with negative temperature to a system with positive temperature if those systems are in contact. It's easier to think in terms of the inverse of the temperature ("coldness" or "thermodynamic beta"): energy flows towards higher coldness, and you no longer have any special cases.

Here's an intuition that might help: Suppose we define β=1/T, the reciprocal of temperature. (See https://en.wikipedia.org/wiki/Thermodynamic_beta.) As a system gets hotter and hotter, T gets bigger and bigger, so β falls closer and closer to zero. If β falls past zero and becomes negative, then T will also be negative.

(Also: If T=0, then β would be undefined/infinity. This corresponds to the fact that absolute zero temperature is impossible. β is arguably a more natural way of thinking about temperature than T is.)

Arithmetic underflow is not a bad explanation.

How you interpret is that it's pop science misinterpretation. Temperature is necessarily defined for systems in equilibrium. Systems with "negative T" aren't in equilibrium hence T isn't strictly defined.

So, what do we mean by neg T? Solutions to Boltzmann's distribution for population inversion (more electrons, say, in an excited state than the ground state).

Usually, the hotter something is, the more excited states are occupied; but in equilibrium there are always more occupied ground states.

So "hotter than any positive T" refers to "negative T"s having more excited states than positive T

It just means that in a negative-temp system the number of states with a given energy gets smaller as energy goes up rather than, as for the vast majority of systems (which have positive temperature), getting larger. That means that energy will flow even more quickly to the positive-temp system from the negative-temp system than it would from any positive-temp system, because each unit of energy transfered enlarges the state space of both systems (rather than enlarging one and shrinking the other so as to simply produce a net increase in the number of states, as is the case for most spontaneous energy flows).

Arithmetic underflow, obviously.

negative temperature is like a burrito