Using scissors or wire snips, cut the wire just behind the damaged area. Make sure the item is unplugged.
Take the razor blade and carefully remove one inch of the outer covering (if one is present) without cutting into the wires themselves. You will want to have roughly an inch of the wires exposed at least.
Identify the wires. In three pronged plugs, the white wire is always the neutral wire. The ground wire is either green or a copper wire with no casing. The hot wire is the last one and is black. Two prong plugs do not have a green wire.
Using wire strippers (or a razor blade), cut ½ inch of the colored casing around the wire off, exposing the bare metal.
Open the replacement plug according to the instructions. Push the white (neutral) wire into the silver pinch point. Tighten the screws and lock the wire in place. The black wire (hot) goes into bronze or copper colored pinch point. The green or copper wire (ground) goes into green pinch point. All wires should be securely screwed.
Close up the replacement plug, ensuring that none of your wires or casings are exposed outside the plug housing. You should give it a tug to make sure none of the wires move or separate from the plug housing.