Why does the author tells the story in flashback?

The choice of the author to tell this short story in flashback is a bold one, which not only heightens and maintains the suspense but also allows the writer to completely subvert the reader's expectations at the very last moment.
At the beginning of the story, we meet the old man pleading with his son to "tell them not to kill me." Our first thoughts, then, are sympathy for the old man, whom we are sure does not deserve to be killed, and also curiosity to know who wants to kill him and why. As the story goes on, we learn, piece by piece, how the old man has ended up in this situation.
The narrative voice shifts points of view, which also adds to the intrigue of the story. According to the old man's perspective, he did kill Don Lupe, but he "had to," and he "had his reasons." This strengthens our sense that the old man is being mistreated, particularly due to the phrases used which evoke pathos—the old man "clung" to "hope" and wants "peace in [his] old age." We feel sympathy for him because he is old and weak—but, of course, this is also exactly what he hopes his captors will do.
Only at the very end of the story, then, do we hear what happened to Don Lupe from the perspective of the unfortunate man's son. Far from killing him by accident, or even killing him humanely, the old man stabbed him with a cattle prod and left him to die in agony. By the time the story concludes, then, with the old man dead, we understand the actions of his captors, and our sympathies have shifted—we no longer believe that the old man is someone to feel sorry for. We feel for his captors instead.