Zelda is the reincarnation of the Goddess that gave her immortality to seal the evil known as demise from hyrule or whatever you want to call it. Link is the chosen hero of the goddess, Fi is an artifical spirit that lives in the skyward sword, which later turns into the master sword.

Zelda gets knocked out of the sky while flying around with Link by a guy called "Girahim" which later is revealed is the weapon of demise sort of like Fi is the master sword.

He doesn't manage to capture zelda though as she is helped by an older woman and is sent off to "cleanse" herself at 3 temples regaining her memory that she had when she was the goddess.

She gets captured but is freed by a woman called "impa" who is of the shiek clan, they finish the cleansing and travel back in time, for Zelda to seal herself to keep Demise sealed as well untill Link can find the Triforce which will allow him to "wish" demise out of excistence.

He does this, however Girahim intervenes, taking Zelda dragging her back in time, and start a ritual to unseal Demise in the old land before he is destroyed in current time (this is all probably really confusing however it is a lot easier to follow if you actually play it.) He succeeds partially before Link intervenes summoning Demise which is transformed from a furry ball with teeth to a ripped dude with flame hair, Link defeats him.

He then says "My hate.. Never perishes! it is born a new in a cycle with no end" "I will rise again!" "those like you.. Those who share the blood of the goddess and the spirit of the hero is eternally bound to this curse" "An incarnation of my hatred shall ever follow your kind, dooming them to wander a blood-soaked sea of darkness for all time!"

Then they go back to the present but not before saying goodbye to Fi and leaving the master sword in the past sealing the remnants of Demise for good (or was it?) and impa also shares she is a product of her own time and needs to stay and guard the Master Sword, and even though ZElda protests she assures her she will see her again. Zelda then gives her an armband before they go through the portal and meet the old lady who guards the temple in the present and they notice she has that same armband Zelda gave impa, before impa says "I told you we would see each other again Zelda" before she starts to vanish and eventually perishes. Then you cut to outside and Zelda telling Link she wants to live down on the ground, and asks Link what he wants to do, he then smiles and they cut to 2 birds flying away the end.