One advantage of having Mr. Anderson tell the story rather than Cecil himself is that it encourages the reader to identify with Mr. Anderson's perspective on the situation and sympathy for Cecil. Cecil seems to try hard to bear up under the terrible treatment he receives from Moose, only "breaking down" the one time that Mr. Anderson sees him crying on the beach. Perhaps if Cecil told the story, he would try to downplay the effects Moose's meanness has on him. However, Mr. Anderson has no motive to tell anything but the unvarnished truth about his concerns. On the other hand, if Cecil were to be totally honest about his treatment, we might suspect him of exaggerating Moose's cruelty. With Mr. Anderson doing the telling, we know how horrible Moose's behavior really is: so horrible that Mr. Anderson is concerned for Cecil's life.
Further, because Mr. Anderson tells the story, we will never know if Cecil was responsible for Moose's death or not. We can only speculate, with Mr. Anderson, as to what really happens that night when Moose falls to his death, when Cecil "goes to bed" very early. If Cecil told the story himself, this intentional ambiguity would disappear, and it would make the end of the story rather less memorable and chilling.