An alligator clip is a small, spring-loaded metal clip that can be used to make temporary connections between two wires or between a wire and the anode or cathode of a device. The clip has one end where a wire is screwed into place whilst the other end can be clipped or unclipped as needed.
Use the wire strippers to strip away an inch or so of plastic covering at the end of the wire you want to attach to the alligator clip. Do this by pinching the wire strippers around the wire just tight enough to cut through the coating and then pulling it away; it will slide easily off the wire.
Thread the exposed end of the wire through the circular hole at the back of the clip.
Unscrew the small screw a few turns and loop the exposed end of the wire around the screw before tightening the screw to hold it in place. If the clip does not have a screw it may be that it requires crimping to attach to the wire. The clip will have two little metal wings sticking out from the sides. Place the exposed end of the wire between flat on the clip between them and use pliers to squeeze the wings over the wire to hold it tight. If it has neither of things simply loop the wire as tightly as possible around the end of the clip and squeeze it flat with the pliers.
Establish a permanent connection using a soldering iron and solder. When the wire is in place push the hot soldering iron gently against the end of the roll of solder, making a small blob of molten solder on the end of the soldering iron. Wipe this over the alligator/wire connection and allow to cool.
Open the jaws of the alligator clip by pinching on the back of the clip and put the clip on to whatever you are trying to connect.
Be careful when working on electrical equipment: make sure the current is switched off before you start work on circuits. Be careful using soldering irons as they become very hot.