Brake light problem

Yup. Break in the ground circuit. You can verify this by turning on the tail lights, then remove the right bulbs and you will see the left ones turn off.

Current wants to go through the left tail light bulb, but it cannot find a path to ground. Instead, it goes through the tail light filament, then through the brake light filament in that same bulb, through the splice, then over to the right bulb's brake light filament, then to ground. Current in the left side is going through three filaments, so each one drops about four volts, so they are dim compared to the right side.

Start with a visual inspection of the wires. You should have sockets with two contacts in the center. Those are for the brake / signal light and the tail light. There will be a third wire attached to the metal outer housing. That is the ground wire. Check it for breaks, and it will end by being bolted to the body sheet metal somewhere in the trunk. That is a good place to find a rusty bolt or the wire is corroded off its terminal.

Keep me updated. The next step is to take a voltage reading. That is real easy to do, ... Not so easy to describe if you don't know how to do it.

Dandy. I'm not going to whine and snivel with your success, but for most wiring problems, you want to find the location of the defect so you can figure out what caused it. A corroded splice won't affect another wire in the harness, but if the wire grounded out because the harness is laying on the sharp edge of a metal bracket, it's only a matter of time before the same thing happens to the next wire. We want to find that before our customer has another problem that could have been avoided.

There isn't much involved in the rear wiring harness. A good suspect is if someone used Scotch-Lok connectors to splice in a trailer wiring harness. Those connectors don't seal out moisture, so corroded wires can be expected.