Originally Posted by BrianX54.4is

To remove the third brake light you first need to remove the spoiler cover (5 torx screws).
This video covers it.
BMW X5 third brake light removal Audio Rear Spoiler Antenna e53 bimmer merchant - YouTube

On the off-chance that the light is ok but there is damaged wiring at the tailgate hinge (as in my case recently) - see this post.