Here's how it works:
* The Setting: The poem starts with a sense of global chaos and disarray. Yeats writes about a "rough beast" slouching towards Bethlehem, suggesting a return to a pre-Christian world.
* The Expectation: The second coming of Christ is usually understood as a time of redemption, a return to order and righteousness.
* The Ironic Twist: Instead of a messianic figure, the poem depicts a monstrous, chaotic force, the "rough beast." The arrival of this "beast" represents the collapse of all order and the descent into primal savagery.
* The "Happy" Element: The poem ends with the line "And what rough beast, its hour come round at last, / Slouches towards Bethlehem to be born?" The fact that the "beast's" hour has come is interpreted as a sense of completion. It's as if the chaos has run its course and the world is ready for a new, albeit terrible, era. This finality can be interpreted as a grim form of resolution.
* The Dark Ironic Twist: The "happy end" is a dark one, as the world is entering a period of ultimate chaos and barbarity.
So, it's a "happy end" in the sense of a definitive conclusion, but the result is anything but happy or positive. The irony lies in the stark contrast between the expected return of Christ and the terrifying reality of the "rough beast."
In short, the poem's "happy end" is a bleak, darkly ironic twist on the concept of the second coming. It's a conclusion, but a terrifying one.