Go to any kickboxing, boxing or MMA gym and they'll tell you to relax and not be stiff when sparring, even hitting the bag or mitts. Being stiff means that your muscles are already contracted or semi-contracted, which means more resistance that will get in the way of how fast you can throw and how fluid you can strike. Being tense/stiff hinders this. Wanderlei was as stiff as a board, as an extreme example. He never had fast hands but he was always a powerful puncher. Dan Henderson too, even Rumble. All of these guys are "stiffs" but have been very effective with their punching power and getting the knockout.
Defensively there are benefits to staying loose and relaxed too. When a punch comes in and you're loose and not already tensed up you'll be able to absorb it more like a sponge than if you were stiff, where the punch (or any strike for that matter) would meet it dead on with full impact because there's little to no give to take some of the impact away. When Conor was eating GNP from Mendes, notice how extremely relaxed he was. By doing this, it made him a lot harder to hurt with those punches and elbows.
Same concept when getting into a car wreck. If you tense up then you risk being injured worse than you'd be if you were relaxed. You go with the momentum/force, you don't want to fight it and meet it head on.