Why the Quick-Shifter?
Since these are called quick-shifters we'll dub slow-shifting the traditional way of shifting gear. When you slow-shift you usually close the throttle, disengage the clutch, act on the gear shifting pedal, engage the clutch and reopen the throttle. Now, this is a time-consuming process (a few hundreds of milliseconds) which might be undesirable in a racing environment where every fraction of a second counts. Enters: the quick-shifter.
How Does it Work?
There are many types of quick-shifters out there and not all use the same methods although they all aim at achieving a common goal: reduce load on the drive-train to safely engage a higher gear without damaging the gearbox. The way this is done is usually by momentarily cutting fuel and/or ignition for a few tens of milliseconds during which the gear is engaged. All of this happens without the need to close the throttle.
Just to give you an idea of how fast a quick-shifter can operate, see this video of a quick-shifting ZX10.
The quick-shifter is usually a sensor reading the position of the shifter, informing a microcontroller of when up-shifting takes place in order for the controller to cut the fuel/ignition. It looks something like this:
Image courtesy of Kawi Forums
A quick-shifter circuit typically looks like this:
Image courtesy of R6ers
In this case you have one cable for each sparkplug which suggests that this unit cuts the ignition when quick-shifting.
Is Quick-Shifting Harmful for the Gearbox?
The general consensus on the internet seems to be that quick-shifting can be less harmful for the gearbox than slow-shifting since the engine power is guaranteed to be lowered when the fuel/ignition are cut. Whereas an unskilled driver manually operating the clutch is more likely to make the mistake of leaving the throttle open during up-shifting.