In the story "The Miracle of the Birds," what is the author arguing?

I would suggest that Abado isn't putting forward a singular central argument in "The Miracle of The Birds." But there are undoubtedly a number of themes and familiar tropes that form an integral part of the literary tradition in which the author's work is best understood.
One theme is that of freedom. Ubaldo, the protagonist, is a poet and troubador who wanders from place to place. He plies his in-demand trade at weddings, funerals, and baptisms. His is a life of freedom: the freedom of the open road and air. He cannot truly settle down as he is also a notorious philanderer, and he is known to leave behind many a broken heart in his wake.
Yet Ubaldo is not truly free. For after sleeping with the Captain's wife he comes perilously close to being killed by her irate, gun-slinging husband. With freedom comes responsibility, we might say; Ubaldo's obvious lack of responsibility for his actions diminishes his freedom.
The birds he inadvertently frees from their cages show the way. They are truly free now that they have been allowed once more to reconnect with their nature as winged creatures. And perhaps by depositing Ubaldo in a convent, a place full of virgins, there is just the faintest suggestion from the author that Ubaldo too needs to get in touch with his true nature: one that doesn't involve seeing women simply as objects of sexual conquest. Rather, he needs to discover a nature which accepts that our freedom is ultimately dependent on respecting that of others.
This relates to the theme of the relationship between men and women in the story. At work here is the age-old trope of the virgin/whore dichotomy. Sabô, the Captain's wife, is presented as little more than a voluptuous, sexually avaricious tramp and the latest notch on Ubaldo's bedpost. The macho culture of which Ubaldo is an enthusiastic devotee is in evidence here. The nuns provide a complete contrast. Ubaldo's life is saved by virgins in an earthly, rather than heavenly, sense.
It is difficult to set forth a particular point of view here. On the one hand it would seem that Abado is unashamedly celebrating macho culture in his attempt to put before us a colorful, picaresque romp. On the other hand, perhaps he could be trying to show us that machismo does have negative consequences: whether it's the serial womanizing of Ubaldo, who ends up in a convent surrounded by nuns; or the Captain's penchant for armed violence, which ends with him turning into a tree.