The Legend of Zelda franchise has a long and detailed mythology, so get ready to learn. Spoilers below.
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The Legend of Zelda: A Link Between Worlds takes place six generations after The Legend of Zelda: A Link to the Past (the third game from the Legend of Zelda franchise). Link, a resident of Hyrule, accidentally witnesses the evil Yuga transforming a young woman into a painting. The woman, Seres, is one of seven sages, who protect the magical Trifoce and guard the kingdom of Hyrule from evil. The Triforce is comprised of three golden triangles, one representing courage, another wisdom, and the last power. It is very religiously symbolic in the Zelda universe.
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Yuga gets away, and Link receives a magical bracelet from a strange merchant in a bunny hood named Ravio. Hearing his story, Zelda, the princess of Hyrule, grants Link with a magical pendant and instructs him to find two more so that he can obtain the Master Sword. When Link encounters Yuga again, the bracelet saves him from being transformed into a painting, and for the rest of the game, Link has the ability to transform into a painting at will. This allows him to travel along walls or through tiny cracks.
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After a battle with Yuga, Link goes through a small crack and finds himself in the kingdom of Lorule. Princess Hilda, Zelda's direct counterpart, instructs Link to help her rescue the sages. Later, it's revealed that she was using Link and Yuga to steal the Triforce of Wisdom from Princess Zelda. Lorule had once had its own Triforce, but it was destroyed, and Hilda is convinced that Hyrule's Triforce will return Lorule to its former glory. However, Yuga betrays her. Link and Yuga have a final battle, and Link is the victor.
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Hilda is still desperate to take the Triforce for Lorule, insisting that she needs it to save her people, but Ravio convinces her not to do it. Link and Zelda return to Hyrule, and decide to use the power of their Triforce to restore Lorule to its former glory.