I'm a bit picture challenged these days but will try to make it clearer.
So you have the humbucker ring sitting upside down on your table, the pickup height screws through the ring's height screw holes and pointed at the ceiling -- in other words, the normal position when you're about to mount a humbucker in the ring.
Then you put the springs on the screws as always. And as always, the springs are longer than the screws. And once again as always, you push each spring down to get at the end of the screws so you can thread it into the ears of the humbucker baseplate.
Instead of futzing with holding the spring and holding the pickup and needing a third hand to turn the screw and thread it into the hole, and getting to a point where you're ready to throw the pickup through a wall as the spring pops off and hits the ceiling and lands God only knows where 50 times in a row, voila! the alligator clip is your 3rd hand.
You push the spring down with one hand like normal, to where about a third to half of the screw is showing bare above the spring, then you put the alligator clip on the screw just above the spring and said clip keeps the spring from flying off. But again, make sure the clip is holding the screw at or near the back of its jaw for max leverage to keep the spring from blasting the clip off.
Then you thread the screws into the baseplate ear holes a few turns, stop, remove the clips, and finish turning the screw to the appropriate height for installation of the ring-mounted pu into your guitar.
You need small gator clips like the ones available at Radio Shack.