Jungian analysis may be a very good way to try and understand fairy tales. In Jung, archetypes are fragments of psychic ideas, which when put together into a story, can help to point to aspects of the Self. In Jung, the unconscious is divided into the collective unconscious and the personal unconscious, where the collective unconscious is genetic and shared by all humans. Both of these include multiple aspects, sharing the concept of the Ego, the Shadow, and the Self. Further, Jung believed that archetypes reside in the collective unconscious. The idea is that these archetypes are unconscious ideas that are pointed to by aspects of fairy tales. Examples of archetypes of the collective unconscious include the wise old man, who appears in many stories as a guiding figure for the hero but always dies or leaves the hero alone before the final act of the story, or the woods, which apparently symbolize an awakening or the start of a journey. Similarly, the villain in many stories represents a darker version or side of the hero, namely, the shadow, or the part of our consciousness of which we are not aware.
One concept which seems to be important is that when a hero goes a journey, in some fashion, they are immersed in a world which is not their own. By the end of the story, they have executed actions which not only make that world familiar and comfortable, but also made their original world better. They have become master of both worlds.
Jung thought that the archetypes of any given fairy tale together point to some greater unconscious concept, a piece of the Self. Fairy tales in Jung are the ultimate way of expressing the collective unconscious physically.