Why? What led me to believe that a book with "a story by" in its title would enclose anything but fiction? Why did I want the tale to be true?
First, the story is told in the first person, which for me generally increases the believability of a story. True, it's a pretty minor reason, but it's a real one.
Second, the narrator uses a straightforward, simple style to tell the story of the fictional character of Elzeard Bouffier. It's not news-style prose, but the ornamentation in the descriptions is subtle and unobtrusive. The reader gets the sense that Giono is telling things like they are/were, and that he himself met this M. Bouffier and watched the forest grow over time.
Third, and more importantly, the tale seems plausible. At the first visit by the narrator, Bouffier goes through a sack of acorns (presumably collected from previously planted oaks) and carefully sorts out one hundred perfect acorns, which he soaks overnight and plants with the aid of a pointed stick the next day as he monitors his sheep. He expects something like ten percent to survive to maturity. Over the course of only a few years, those acorns add up to a thousand trees. By the time of the final visit decades after the first, the forest is thriving. Both the act and the results of planting the acorns seem like they really could happen.
Fourth and finally, I found I really wanted for a person as selflessly dedicated to a simple but world-changing task as Elzeard Bouffier to exist. The narrator quite rightly calls him "one of God's athletes" near the close of the story. The foreward points out the resemblance between Bouffier and Saint Francis - both humble men in close communion with the earth. Can't we have such a saint in the modern age? Why shouldn't such people exist?
According to both foreword and afterwords, Giono's tale has inspired a lot of tree-planting worldwide. These appended stories are nearly as remarkable in their end results as the central one: planting the right trees in the right areas will clean the air, bring groundwater up to the surface, and cause land that was harsh and barren to come back to life. The differences between the tales are that the real-world stories were universally the feats of many people working together rather than individuals working in insulation... and that the real-world tales are true.