While it's clear that outdoor cats enjoy the freedom of being outside, whether or not they are "happier" is impossible to tell, and it is certainly true that they are not healthier.
Outdoor cats face risks that indoor cats do not. They risk suffering at the hands of cruel humans, being torn apart by wild animals and dogs, being run over by cars, and being subjected to disease, parasites, and weather. Outdoor cats are responsible for millions of bird and small animal deaths, and cats that aren't spayed or neutered add to the overpopulation problem.
Indoor cats are healthier, and they can be just as happy. Make sure that your indoor cat has plenty to do–provide toys, cat grass, scratching posts, and even better, cat trees on which they can satisfy their need to climb. Providing them with a window perch gives them an outside view and sunlight in which cats love to bask.
I cannot stress how dangerous it is to allow your cat to go outside (not to mention inconsiderate of your neighbors who have to deal with your cat relieving itself in their sandboxes or spraying on their front doors.) But if you must let your cat go outdoors, be responsible. Make sure your cat is vaccinated and protected against parasites like fleas and ticks. Neuter or spay. Fit your cat with a microchip and a breakaway, reflective collar, and include a bell to warn wildlife of your cat's approach.