The way the drop-D eXtender works is it turns the tuning peg a fraction to detune it to the note you want, typically D.
It does exactly the same as you would do if you grabbed the tuning key and turned it.
The only difference is it has a lever to do it for you and allows you to set the turn precisely via a thumscrew to set the stop-point of the rotation.
This pic shows two of them and you can see that the levers are pulled and the tuning key has dropped backwards maybe 1/16 of a turn as the whole tuner chassis has rotated.
Looking behind, the way it works is that the whole original tuner is a floating assemble which pivots around the tuning peg and sits on top of a backplate which is fixed to the headstock.
Pulling the lever allows it to rotate the requisite amount.
It's great if you want to easily flip between the two, but if you're going to permanently play in drop-D then there's no point in having it.