
The movie begins with the princesses Elsa and Anna as children (they seem to be about 8 and 5, respectively). As is evident from the opening scenes, Elsa has cryokinetic powers and can control ice and snow. In the opening scenes, she is entirely at ease with her powers, and her gifts bring her and her sister great joy and love. Children are usually whole, with no distinction between the Light and the Shadow; they simply ARE.

However, after Anna almost dies when Elsa accidentally blasts her with ice, her parents are understandably terrified. They take Elsa and Anna both to the trolls, whose magic saves Anna from death. The troll elder tells Elsa and her parents that her gift has great beauty, but also can be very dangerous (sometimes I wonder how different the movie would be if the troll elder had told Elsa instead that her gift is dangerous but also beautiful!). The King determines that the best thing to do is to conceal her gift and to keep it hidden, in part by isolating Elsa as well.
Now, Elsa has learned that there is a part of her that is dangerous and needs to be rejected if she is to be a "good girl". She repeats the mantra, "Conceal it, don't feel it. Don't let it show". Since her powers are triggered by emotions, Elsa struggles to keep her emotions under control and to feel nothing at all. She even dons a pair of gloves as a physical reminder to repress her powers. Her powers over ice have become her Shadow. Outwardly, she projects the image of the perfectly prim and proper princess, well-mannered and dignified. Yet the process of cutting off this piece of herself costs her the ability to express love towards others -- most tragically, she loses her connection to Anna, who also suffers from isolation and neglect.
Yet the Shadow isn't so easily rejected. We see that externally she tries to be calm and reserved, in the privacy of her room, her Shadow tends to manifest unexpectedly. She has only to put her hands on the windowsill and it inadvertently frosts over. When she loses her parents, she clearly cannot contain her grief, and we see her bedroom covered in ice. Likewise, the more the Shadow is repressed, the more likely it is to rear its head and take over, as it does after her coronation, when her uncontrollable terror plunges her entire queendom into a devastating winter (setting into motion the main plot of the story). In front of her entire queendom, her dignified persona comes crashing down as they realize that she is anything but.
In the song, "Let It Go", Elsa finally begins to embrace her magic. For the first time, she doesn't need to repress her powers, and quickly discovers that it is incredibly liberating to be able to, well, "let it go." She can express her rage, frustration, exhilaration, and defiance freely at last. Similarly, when we first indulge in liberating the Shadow, we find ourselves capable of so much more than we could possibly have dreamed of (Elsa, it turns out, is quite the architect, fashion designer, and battle mage!).
Yet she's still not fully in control -- at one point, she almost lets the Shadow overtake her entirely. She nearly kills two men (albeit in self-defense), and her ice palace begins projecting spikes aimed towards her (representing how it almost destroys herself).

The true catalyst for her acceptance of the Shadow, however, winds up coming from Anna. Throughout the movie, Anna has been discovering the terrible effects of her sister's magic, and it very nearly kills her. Yet she dramatically chooses to sacrifice herself to save her sister's life -- in this, Anna represents unconditional love, who accepts Elsa for both her Light and her Shadow, with full knowledge of both. It is this realization that shows Elsa the secret to her Shadow ("Love will thaw!"), and she finally truly accepts her Shadow as part of herself. In the final scene, we see Elsa, fully in control of her Shadow, using magic borne out of love (for herself, for her kingdom, for her sister) to make an ice skating rink in the castle courtyard to the delight of her subjects. Having accepted herself, good and bad, she can be the dignified queen she always wanted to be -- but with the capacity for levity and joy she lacked before. She has found the gifts in her Shadow, and she can choose to use it for positive, rather than destructive, ends.