AKC is a registry that keeps records of pedigrees and breedings. This is not to say that they couldn't be defrauded by a dishonest breeder, but if the dog is from an ethical breeder, you can have faith in the accuracy of the pedigree, which is important if you consider breeding the dog. One step is to always study the pedigrees of the two that are being considered. Recently, they have begun to use DNA as a means of positively identifying dogs and their pedigrees. The good breeders are already using this quite extensively. I doubt if the BYBs, even if they have AKC registered dogs, will go to the extra expense. Although AKC registration does not guarantee that the dog is top quality, has the temperament you seek, or conforms to the breed standard, you can bet that the very best ones in a breed are registered. It's step one, but you still have to evaluate the dog itself. Unfortunately, less than ethical breeders will try to use that as a selling point, top notch breeders assume you know that their dogs will be registered and don't make a point of if.
AKC also sanctions various kinds of competitions, not just conformation shows. There are also tracking trials, obedience, rally, junior showmanship and other kinds of competitions that they sanction.
Note to Stulisa42 about conformation shows: You are mistaken, it's much more than about appearance. The dog is also judged on structure by observing its movement and through hands on examination, and its temperament is also judged. A dog that does not display a good temperament in the ring does not go far at all. The whole purpose of conformation is to identify the dogs which best exemplify the standard of that breed. I've you've never actually been to a dog show, you should attend one and talk to the owners and handlers. I think it would be quite informative.
AKC is also involved in legislative issues, such as BSL, which may affect your right to even own a dog, in educational efforts, in supporting rescue organizations, and many other aspects of the dog world.
In terms of breeding, having an AKC registered dog does not automatically make it a good candidate for breeding. You put your dog up against the standard in conformation showing, and that is the way to identify the ones that are most likely to contribute to the improvement of the breed. Then you might be in a position to consider a breeding, but there would still be a lot of work and thought that would go into deciding which two dogs should be mated. But to go back to the first point, if you have pedigrees that you can trust of both dogs (the registry function of AKC), then you're in a position to make much more informed decisions about breeding than otherwise.