the turn signal switch is an interupter so to speak, the power to the tail lights will run from the TS switch, when you activat the turn signal, say the left one, it redirects the current for the left light through the flasher, the flasher is a bi-metalic switch that breaks current or the circuit when it heats up then reconnects when it cools hens the flashing signal light.
You need to find the power wire that's hot(has power) when the key is on for your tail lights and hook it to the red wire on your TS switch, the other 12 volt lead on your schematic is to power the flasher when the TS lever is moved to signal a turn, then hook the orange and brown wire to thier corosponding tail lights, should fix ya up.
You should get a factory wiring diagram for your cart, it will help to locate a good 12V source, be sure it's "12" volts, voltage reducers are cheaper than wiring harnesses and electronics
