The main idea for the use of the black card is to capture multiple sets of fireworks into one image.
Basically, you use long exposures or Bulb mode if your camera allows it.
In Bulb mode you control the exposure time manually by pressing the shutter button on your camera (or remote if you have one) to open up the shutter and let go to close it.
The black card comes in when you want to take two sets of fireworks, or more if you want, which go off at an interval into one single image.
Take the first set by opening up the shutter, cover the lens with the black card after the first set is complete without closing the shutter, remove the black card when the 2nd set of fireworks goes off.
Keep repeating it to get multiple fireworks into one frame and close the shutter when it's done.
Just don't get too many of them in one shot though.
When to cover the lens and how long an exposure time you need depends on what you want out of the shots.
Takes a little practice.