The breaking plier is a special stained glass tool that has smooth jaws that meet at the tip of the pliers. This enables the tool to reach over the top and bottom of the glass with only the tip coming into contact with the glass exactly against the score line.
The plier handles are held at a right angle to the score line. The edge of the glass needs to be close enough to the score line in order to use this tool, as the tip of the jaw needs to be against the score line. It is used in lieu of your hands when the piece being broken off is too narrow to be comfortably grasped by hand. When bending the glass, the top jaw comes down flat against the surface of the glass (that's the reason for the smooth jaw) and as the bending pressure is applied, lateral/pulling pressure is applied at the same time. This tool can also be used to groze the glass by carefully nibbling away the edge.
If the score line is further than 20mm away from the edge, cut running pliers are the appropriate tool to use.