Why did the author use the fairy tale structure?

I think there is a certain amount of irony in Lawrence’s choice, if we understand the story as being about the inversion of parent and child roles. The fairy tale is a story meant for children, with a clear moral. Lawrence’s story, on the other hand, is not meant for children, and its message, while seeming to be about greed, is actually more ambiguous. The materialism of Paul’s mother is a symptom of a larger spiritual “lack,” an emptiness that ultimately effects her ability to love Paul—in a way, she is not mature enough to be a parent. Paul, for his part, secretly assumes the responsibilities of an adult in winning the money, even though he is not really aware of how he does it. Paul’s mysterious “being sure” about horses stands in counterpoint to the ambiguity of the love relationships in his own family. 
Lawrence’s use of the fairy tale foregrounds the fantastic elements of the story while at the same time commenting on the nature of Paul’s childhood. It’s not so much that Lawrence is using the form to convey a moral as it is that he is trying to call attention to the fact that Paul’s life is anything but a fairy tale when it comes to emotional experience.

D.H. Lawrence wanted the moral of his story to be clear, so he used a simple fairy tale structure. He aimed to show that the absence of love kills and that no amount of money can fill the hole left when love is absent.
In this story, Paul's mother is a simple representation of a person who, for whatever reason, is incapable of loving. She is a flat, rather than a fully developed, character typical in a fairy tale. She attempts to fill her inner emptiness with money; but as with any addiction, the more she gets of her "drug," the more she wants.
Paul does not understand this and thinks he can earn her love by winning money for her. That he can do it in a supernatural, fairy-tale way by riding his rocking horse to learn which horse will win an upcoming race emphasizes that the problem is with the mother. Paul wins large sums, but they fail to satisfy his mother. If Paul had simply been an ordinary child who tried to earn his mother's love by obtaining a little extra money and giving it to her, we might not understand the depth of the mother's lack. If Paul had not died at the end trying to please her, we might not fully realize the devastating effect on a child of not being loved.