Analysis of a NOT gate circuit
The description is simplified. 

When you close the switch, if the resistor values are chosen appropriately, the NPN BJT will be saturated, so Vce will be reduced to some tens of mV typically. 

With tens (or hundreds) of mV across the LED the current through the LED will be very, very small (but not quite zero) and thus no (perceived) visible light will be emitted.