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.)