Once I had connected all the ESC's and mounted all the arms into place, I began the process of sorting out the wiring mess in the middle. I started by cutting all the power wires and soldering them into the distribution board (the bottom plate). Unfortunately, the specific batch of distribution boards that I received were missing a solder pad for one of the ESC's. This wasn't a big deal because I was able to just branch off the other solder pads.
Next, I took all of the signal wires from the ESC's and wrapped them up with a few zip ties - mainly to keep them from tangling up.