Fairy tales tend not to narrate what happens to you or me. They tell us about That Person Over There Who Is Often Royalty (Or Soon To Become Royalty). Using the third person narrative voice is the most effective way to tell us that Person A was orphaned and is bummed about it, while Person B sees through Person's A disguise to notice that they're actually kind of hot, and sometimes we get a glimpse of Person C's thoughts as she whips up yet another miraculously beautiful dress to bring the two lovebirds together.
For instance, in "Little Red Cap" we get to see the wolf's predatory thoughts: "The wolf thought to himself, This tender young thing is a juicy morsel. She'll taste even better than the old woman. You've got to be real crafty if you want to catch them both" (Little Red Cap.93). But then we see Little Red Cap's thoughts as she enters the cottage: "She was puzzled when she found the door open, and as she entered the room, it seemed so strange inside that she thought, Oh, my God, how frightened I feel today, and usually I like to be at Grandmother's" (Little Red Cap.95). These exchanges help us see that the wolf, unlike other fairy-tale animals, is definitely not a friendly helper figure, even if we do get a chance to relate to his thoughts.
The use of the third person also gives us enough narrative distance to avoid leaking empathy every time someone is killed in fairy tales ("Nameless Suitors," we feel for ya). At the same time, fairy tales manage to be pretty personal. Who doesn't feel bad for orphans and mistreated youngest siblings? Getting the occasional look at their thoughts also helps convince us that we're rooting for the right person. Good and evil tend to be pretty clearly delineated in fairy tales, and the "omniscient" part of the "third person omniscient" definitely helps with that.
Okay, there are two tales that appear in the first person: "The Tale About the Land of Cockaigne" and "A Tall Tale from Ditmarsh" which are nonsense tales and don't really fit the overall pattern, but we thought we'd mention them anyway. What's with the weird change of pace? And what about those dialogue tales? We'll leave those questions up to you.