THat is the turn sig switch. It's easy to think of it as plumbing. 12v is the water, the switches are the valves. Where the line is labeled "From brake switch" thats 12v coming in from the brake switch. The graphic shows the turn sig centered, then right turn sig, then left turn sig. You can see how the pads move over the different contacts and connect different wires together. THis is how the turn sig switch allows the brake lights to come on. If your foot is on the brake, and the left turn sig is on, then the left light will blink, and the right will stay on steady. ANd vise versa.
Wiring is not something you can tear into. You will need to slow down and understand the circuit, and see where you are losing the connection. For now, look at the curved connector going to the steering column. Make sure there are no pins pushed out backwards. If you have 12v to one side of the brake light switch, and 12v on the other side of the brake light switch when the pedel is pushed, then you are good to go "from that point". THe next connection point is the curved connector. THen the turn sig switch under the steering wheel.
I know how frustrating it can be. But we can fix it. Let us know what you find next.
Where are you located? THere may be someone near you that can help. If you like, (and some perfer not to) you can edit your profile and put your location up so everyone can see where you are at...