the dangling plugs from the taillight are for the LEDs..... those plugs are supposed to replace the bulbs from the car's stock harness.
i don't know the type if taillights you bought without actually seeing them in person and testing each plugs.
from your picture.... the upper hanging plug with 3 wires is supposed to replace the brake/tail bulb --> yellow socket... while the 2 wire one replaces the turn signal bulb --> white socket.
LEDs are polarized.. so you may need the flip the plug 180 degrees to get it working.