The falling action and resolution of Old Yeller happen after the climax (which is the moment that Travis is forced to shoot Old Yeller in order to prevent him from suffering through rabies).
Therefore, the falling action involves the emotional turmoil that Travis goes through as a result of Old Yeller's death. In the falling action, Travis chronicles his indifference to absolutely everything, from the fact that he still has one of Old Yeller's puppies to the fact that his father finally bought him the horse he always wanted. Even after a good talk from Papa about not letting the bad and unfair things about life take over the good things, Travis still isn't sure.
What I mean is, things like that happen. They may seem might cruel and unfair, but that's how life is a part of the time. But that isn't the only way life is. A part of the time, it's mighty good. And a man can't afford to waste all the good part, worrying about the bad parts. That makes it all bad.
The resolution happens when Travis finally begins to appreciate the good things in life again (ironically due to a stolen piece of cornbread). The speckled puppy truly does act like a young Old Yeller, and Travis begins to notice this. It makes him happy. Travis rides his new horse. That makes him happy, too. Further, the speckled puppy's antics make Travis both laugh and cry allowing him to let out his grief for Old Yeller. Life goes on.