Transistor as switch and voltage drop
Your understanding is pretty much correct.

When the transistor is turned on it is driven into saturation and becomes a very low resistance in parallel with the LED. This pulls the voltage down towards zero - maybe 0.2 V and this isn't enough to light the LED.



Figure 1. LED I versus V curves.

If you pick the colour of your LED and have a look at the graph you can see that at 0.2 V virtually no current will go through the LED.



Your example circuit is using the transistor to "shunt" the current away from the LED. This is a bit wasteful of energy as it draws more current when the LED is off than when it's on. The more usual arrangement is to put the LED in series with the resistor.