Turn off the power to the cable that is damaged. Whether this is done through a breaker box or by disconnecting a battery, this is an essential step for safety.
Identify the damaged section. Determine whether there is only one wire or multiple wires within a single sheath. Find out whether the copper core of the wire is still intact, and check to see whether the metal of the wire is fused to another wire. When a short occurs, huge amounts of electricity can be forced to flow, actually welding two wires together.
Cut the wire before and after the damaged section and discard. Cut off two inches of sheathing on the two exposed ends of the non-damaged cable.
Select an appropriate length of replacement cable and cut off two inches of sheathing from each end, making sure the gauge of the replacement cable is the same as the one that was damaged. It would be helpful if the color pattern of the interior wires of the replacement cable were the same as the cable you are repairing.
Expose one inch of copper wire core from the ends of each individual wire inside the cables. There should be four ends that you will splice: two on the replacement section and two from the section that was removed. Remove the shielding using the wire cutters, taking care not to cut the wire itself.
Splice the wires together by intertwining the ends, taking care to match similar colors. Do this on both sides of the section to be repaired.
Cover all exposed copper wires where spliced with at least three layers of electrical tape. Ensure that all exposed metal wiring is covered tightly with electrical tape, then cover them all with another layer to help to tidy the repaired section.
Turn on the power again, and test for proper functionality.