Is it real carbon fiber or that fake stuff they have for car interiors? It depends alot on what its made of, theres different ways of making it and what to make it from IE: aluminum, glass, kevlar, etc.. Also different polymers that can be used to make it. You should ask the manufacturer about what its made of and what process to make it was used to be for sure. Theres afew variables that decides if its good and for what purpose. But it probably is just for looks, as it would be more expensive to do a proper job for strength and impact resistance, UV degrades it also so just something to in to mind.
The patches are designed to add stiffness to the tail in the area of the fins and alter the flex pattern in the back 16-18 inches of the board (assuming it's real carbon fiber). What this does for your surfing is, well, up for debate.
Mostly it's probably just because they look "cool".
Yea... carbon fiber adds stiffness. It's about triple the stiffness of fiberglass. It's true that carbon fiber's impact resistance is low, but it's stiffness helps prevent heel dents, because a heel dent is more of a pressure strain (deformation) then a point of impact failure, like a fracture/ding. So... aside from looking cool (which they do, IMO), if it is true carbon fiber it will stiffen the tail and help prevent pressure dents from your heel.
I think there's better material to use than carbon fiber for tail patches. The reality is, ANY reinforcement in the tail can help prevent heel dents, even an extra glass patch, which has been done ever since surfboards have been made of foam and fiberglass. So why use something as expensive and stiff as carbon fiber? When I'm foiling a board, two things are most important... volume distribution and flex. I design HPSBs to flex like a bow... thin and flexible at the tips, and progressively thicker moving toward the thick point. This allows the tail of the board to flex in such a way that the tail rocker increases as greater load is applied with the back foot on a turn, then sort of "twangs" back coming out of the turn. Added flex during the turn, and the resulting increase in tail rocker tightens the radius of the turn, and helps carry speed through and out of the turn as the tail snaps back into normal position.
So for me, tail patch material should be chosen based on it's ability to resist denting by acting as a bulker in the skin, but still be able to flex. Something light and hard, that has good directional flex and strength, good flex return, easy to cut (which carbon fiber is not), easy to laminate (which carbon fiber is not), not expensive (which carbon fiber is), and looks good.
Like I said, I got 5 boards all with this CF. Would a small patch of CF even increase the stiffness that much? Its not like the entire tail is CF. Would I really notice a difference when surfing compared to a "non" CF Board?