Negative temperature means that the system loses entropy if you put more energy in. This means that negative temp is hotter than any positive temperature, in the sense that if you put two systems in thermal contact, heat will flow from negative to positive.
In the continuum of temperatures, positive absolute zero is the coldest, going up to positive infinity Kelvins. Then minus infinity Kelvins is just slightly hotter than positive infinity. Then it keeps going up to negative absolute zero, the hottest temperature.
These infinities keep cropping up because the reciprocal of temperature is actually the more fundamental physical quantity.
Negative temperature means that the system loses entropy if you put more energy in.
Of course that's basically the same as the "you travel backwards in time if you move faster than light" in the sense that these are things the math does, if you put in these totally unphysical values, while the same set of rules that give this predictions say you can't reach these values.