The best way (high quality of job, pride in workmanship) is to:
- Pick up an assortment of different diameters of heat shrink tubing.
- Remove the entire harness from the bike. Do NOT skip this step. If you try to cut corners here the results will be bad. (No Whining!)
- Cut the sheathing back ten inches from the damage area.
- Purchase up a few feet of the same gauge of red braided wire.
- You are going to be soldering in replacement sections of wire. You will want at least a 1/2" of copper overlap, preferably 3/4" or more (and that's why we had you purchase extra wire in the step above)
- You are going to have to really think about the heat shrink tubing before you start the repair.
- You will want a separate heat shrink on each wire and a larger heat shrink on the harness as an assembly. That will require you to think about each step in the process and think about four steps ahead so the heat shrink is in the right place at the right time.
An alternative method is to use these crimp on connectors:
They'll work, maybe. Until they don't. I'm not a fan.
Good luck with it.