Step 1: Prepare the Wires
Cut away any damage in the wires. Then, using wire strippers, strip about 1 inch of insulation off each wire.
Before continuing, make sure to slide appropriately sized heat shrink tubing over 1 side of each wire being repaired. You want to use enough heat shrink tubing to completely cover the copper that has been exposed. This prevents electrical shorts.
Step 2: Join the wires
Twist the exposed wires around its mate.
Notice the heat shrink tubing. It is about twice the length of the twisted copper pairs.
Step 3: Solder the Wire Joints
Using a soldering iron, cover each wire joint completely with solder. As the copper wire heats up, the solder will be sucked into the strands of the wire, creating a physically solid and electrically stable connection.
Let the soldered wire joints cool before continuing.
Step 4: Shrink the Heatshrink
When the solder is cool, slide the heat shrink tubing over the solder joints. With a match, lighter, or heat gun, apply heat to the heat shrink tubing. It will, as its name implies, shrink and become snug around the wire. This creates a non-conductive barrier around the exposed solder joints. In the repair pictured above, I applied another larger piece of heat shrink tubing around the two joints to make the cable singular again, and without a split.
Thanks for reading!