The number one reason is letting your pet have its independence. When you are away from home, your pet can enjoy the outdoors, yet be inside if the weathers poor or if they need a safe place.
A pet door also allows your pet to use the “outdoor” bathroom rather than making a mess inside, or for cats, this can eliminate dealing with kitty litter. We expect your house will have less “pet” smell when they spend part of the day outside. By reducing or eliminating the need for carpet cleaning and kitty litter you can usually recover the cost of your pet door in less than a year.
Many pet doors also have direction control. If you want to keep your pet inside for the night, you can set the door for incoming only, and once they come in, the pet door will not let them back out until you change the setting.
Rather than constantly being bothered to let your pet in and out, a good pet door puts you back in full control and makes for a happier well-balanced pet!