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Secure the jaws of a wire stripper 1 inch (2.5cm) from the end of one of the wires you're splicing together. Squeeze the handles together firmly together and pull the jaws toward the end of the wire to remove the insulation. Repeat the process on the end of the other wire you're splicing as well. You can get wire strippers from your local hardware store. If you don't have a wire stripper, you can also slice through the insulation with a utility knife. Just be careful not to cut through the actual wire inside. If you accidentally break strands off of a stranded wire, then the wire could cause a fuse to blow. Cut any remaining strands on the wire and try stripping it again. Get heat-shrink tubing that's a gauge larger than the wire you're using so you can easily slide it on. Cut a piece of tubing that's at least 2 inches (5.1cm) long so it's able to cover the splice and some of the insulation later on. Slide the heat-shrink tubing onto one of the wires and move it at least 1 foot (30cm) away from the exposed end. You can buy heat-shrink tubing from your local hardware store or online. Avoid using heat-shrink tubing that's too large for the wire since you may not be able to secure it completely. Don't keep the heat-shrink tubing near the area you're soldering since it could shrink from the heat of your soldering iron. Line up the centers of the exposed wires so they form an X-shape. Bend one of the wires down to twist it around the other wire as tightly as you can so it has a firm connection. Make sure the end of the wire doesn't stick up or point away from the splice or else you won't have as firm of a connection. Repeat the process with the other wire so your splice looks even on both sides. Alligator clips are small metal grips that work well for holding wires in place without them moving around. Place the alligator clips vertically on a flat work surface so the jaws face up. Secure each of the wires into 1 alligator clip so the splice is supported off the work surface between them. You can get alligator clips from your local hardware store. Make sure you work in a well-ventilated space since the fumes from the soldering iron can be harmful. Use a scrap piece of metal or a non-flammable material under the alligator clips to catch any solder spills. Rosin flux is a compound that helps clean the wires and allows the solder to stick to them. Put a bead-sized amount of rosin flux on your finger and rub it over the exposed wires. Try to coat the wires as evenly as possible so there's a thin layer of flux on them. Wipe any excess flux off of the wires with your finger or a paper towel. You can buy rosin flux from your local hardware store.

Summary:
Strip 1in (2.5cm) of the insulation off the end of each wire. Slide a piece of heat-shrink tubing onto one of the wires. Twist the ends of the wires together to combine them. Clamp the wires in alligator clips to keep them off of your work surface. Put rosin flux on the spliced wire to help the solder adhere better.