How does magic in "The Handsomest Drowned Man in the World" inspire villagers to change their lives?

The "magic" you refer to has to do with the fantasy elements of the story. The tremendous size of the corpse, his great beauty, the fact that the corpse apparently has travelled a long way in the sea before washing up on the beach of the village, the fact that the corpse has not decomposed at all after all that time in the water, and then additional days being prepared for disposal by the women—these things suggest that the corpse must mean something more than what it is. For the women in the village, the corpse represents a kind of ideal man, a standard of virility that their husbands cannot match. More than that, the corpse reinvigorates the village—as a result of Estaban's brief sojourn with them, the houses become bigger and stronger, more flowers grow, and the people take a greater pride in being from the village. Perhaps the most "magical" thing in the story is how something dead could confer such life on a place!

How does magic in "The Handsomest Drowned Man in the World" inspire villagers to change their lives?

Is your question referring to magic realism? This is the literary style that combines incredible events with realistic details and relates them all in a matter-of-fact tone. This style emerged from Latin America around the middle of the twentieth century and has gained world-wide popularity. Magic realism blurs the lines that usually separate what seems real from what seems imagined or fantastic.

There are certainly many elements of magic realism in the description of "Esteban", or the drowned man, which certainly serve to make him as a character have the massive impact he does on the village. These would include the massive size and beauty of the man. Note how the narrator describes him using magic realism:

Not only was he the tallest, strongest, most virile, and best-built man they had ever seen, but even though they were looking at him there was no room for him in their imagination.

The clearly mythical status that this man has, described using magic realism, therefore explains the way that the villagers feel strangely attracted to him, claim him for their own, and then are able to transform their lives because of the experience of having the man in their presence. It is this "magic" that perhaps you are referring to in your question.

In "The Handsomest Drowned Man in the World," how does the drowned man transform the villagers and their village?

In Gabriel Garcia Marquez's short story, "The Handsomest Drowned Man In The World," we see the people of the village expand themselves (mentally and physically) in order to take in the enormity of the drowned man—his physical self and their idea of him.

One way to interpret this story is through the lens of storytelling itself. The drowned man is, after all, only a dead body. The entirety of his identity—as Esteban, who is beautiful, strong, hard-working, humble, obliging, and sincere—is all constructed by the people of the village as a story (or rather, a series of evolving stories). As the women clean and dress the drowned man's body, they imagine the person he might have been if he had lived in their village—this is notable because it has no chance of being true. It is pure story. They imagine the amazing things he would've been capable of, his huge house and his happy wife. During their fantasies, they compare this story of Esteban to the men of the village:

They secretly compared him to their own men, thinking that for all their lives theirs were incapable of doing what he could do in one night, and they ended up dismissing them deep in their hearts as the weakest, meanest and most useless creatures on earth.

After idealizing Esteban in their story (and subsequently comparing that perfect hero to their own flawed human men), they take the next step in storytelling—giving their hero a conflict to rail against, something to show his pain and his flaws, so that they can empathize with him. They imagine how hard it must have been for him to have such a huge body, and how people must have treated him differently, walked on egg shells around him, and talked behind his back.

He looked so forever dead, so defenseless, so much like their men that the first furrows of tears opened in their hearts... the drowned man was becoming all the more Esteban for them, and so they wept so much, for he was the most destitute, most peaceful, and most obliging man on earth, poor Esteban.

Here, their ability to round out and empathize with the character in the story they have created allows them to see how he is "so much like their men." Story allows us to connect our empathy not only with characters, but then extend that empathy back to the people we interact with in life.

And after the men return and begin to prepare the drowned man's body for his funeral procession, they are clued into the story the women have begun to tell. At first they resist it, but once they are invited in, they begin to add their own parts to the story— about Esteban's shame and his hope for discretion, which end up reflecting their own feelings, as all stories do.

There was so much truth in his manner that even the most mistrustful men, the ones who felt the bitterness of endless nights at sea fearing that their women would tire of dreaming about them and begin to dream of drowned men, even they and others who were harder still shuddered in the marrow of their bones at Esteban's sincerity.

After the people of the village have told this story together, they bring people from other villages to share in the story, and then after they let Esteban go, they vow to change the ways their lives were small in order to fit this new story. They strengthen their empathy, their openness, their capacity to work together, their welcomeness toward newcomers, and they make the most of their lives. They make the houses bigger to fit his memory, paint the houses bright colors to celebrate him, and dig for springs among the stones just as they imagined he would've if he lived there. This shows the power of the stories we tell, to change the way we act in real life, to improve ourselves, our communities, and our environment. And in the last movement, in which the villagers imagine people far across the sea seeing and smelling all the flowers in their village, there is a hint of the capacity of stories to spread, to influence others, and bring them into the fold of change.

In "The Handsomest Drowned Man in the World," how does the drowned man transform the villagers and their village?

It is clear that the presence of Esteban, or the drowned man, has a massive impact on the life of the village and its inhabitants. Let us remember how the village is described at the beginning of the story:

The village was made up on only twenty-odd wooden houses that had stone courtyards with no flowers and which were spread about on the end of a desertlike cape. There was so little land that mothers always went about with the fear that the wind would carry off their children...

The village is clearly described in very bleak and drab terms, with the greyness and the lack of colour emphasised. Yet, by the end of the story, as the villagers celebrate the funeral of "Esteban", note how they plan to change their lives:

But they also knew that everything would be different from then on, that their houses would have wider doors, higher ceilings, and stronger floors so that Esteban's memory could go everywhere without bumping into beams... they were going to paint their house fronts gay colours to make Esteban's memory eternal and they were going to break their backs digging for springs among the stones and planting flowers on the cliffs...

It is clear that Esteban has helped the villagers realise that they can transcend their own limitations and transform their own lives. His presence has given them the inspiration and the self-belief necessary to see that they can enact massive changes in their lives, in spite of the challenges, and transform their way of life.