Why does NMOS threshold voltage drop when temperature increases?

NMOS threshold voltage drops because the energy gap of silicon reduces with temperature. When the bandgap becomes smaller, less voltage is needed to form the channel. For example, as you heat up a transistor, it needs less gate voltage to turn on.

it’s a combined effect of bandgap narrowing and higher carrier generation. In simple words, when temperature goes up, the material becomes more conductive by itself, so the MOSFET needs less gate push to switch on. That’s why the threshold voltage keeps dropping with temperature. 

It’s because of the intrinsic carrier concentration. As temperature rises, more carriers are generated in the substrate. That means the device already has some “ready-made” carriers, so you don’t need as much voltage at the gate to create the channel. This lowers the threshold voltage.