Obedience, whether the sport or practical, helps you gain trust in your dog. Dog owners have a responsibility to make sure that their dogs are well behaved members of our society. A dog that is taught to pay attention and to follow certain commands is usually more secure, and certainly much safer than a dog that is not well schooled. The goal is to give the dog confidence that you know what is going on so that the dog can rely on you to give it direction. That helps reduce anxiety in both dog and owner.
Training can improve your relationship with your dog. While obedience is a competitive sport there is also practical obedience which is not much different than teaching basic manners and communication. If you are uncomfortable with the formality and, yes, rigidity, of formal obedience you can still become a happy team in practical obedience.