The narrator is contemplating the metaphysics of happiness. He has decided that there is really only one thing—and one thing alone—which has the power to bring genuine happiness to a person. That one thing is the ability to look back at the end of the day with an honest and faithful assertion that that day belonged fully to him.
Owning the day means finding the joy within it regardless of the state of the weather. Owning the day means expanding beyond the uncontrollable fate of the climate to seek, find and own the joy of the day regardless of any other fate outside his control.
Happiness belongs to anyone who has so successfully completely disconnected control of his life from uncontrollable mechanisms of fate that he can honestly tell those fates to do their worst tomorrow since on this day, he lived his life under his terms.
The poem ends with one more bit of terms of the condition for being happy. One must be willing to admit that if even God in His heavens can no longer control what happened in the past, what chance does a man have? To own today not only means opening yourself up to the vagaries of tomorrow, but also not allowing the failures of the past to influence the present.