This was found within the initial game:
To see this, all you had to do was wait at the title screen. (Simply do not press the Start button on the title screen. The music will get a bit more up-beat and this will scroll onto the screen for a bit.)
You could say that the title at the top of this screen represented the name of the game, but I always thought of this text as the simplified, summarized version of the legend.
The instruction manual for the first game introduced a slightly longer backstory, noting Link saved a woman named Impa who told Link this information.
A long, long time ago the World was in an age of Chaos. In the middle of this chaos, in a little kingdom in the land of Hyrule, a legend was being handed down from generation to generation, the legend of the 'Triforce'; golden triangles possessing mystical powers.
One day, an evil army attacked this peaceful little kingdom and stole the Triforce of Power. This army was led by Gannon, the powerful Prince of Darkness who sought to plunge the World into fear and darkness under his rule. Fearing his wicked rule, Zelda, the princess of this kingdom, split up the Triforce of Wisdom into eight fragments and hid them throughout the realm to save the last remaining Triforce from the clutches of the evil Ganon. At the same time, she commanded her most trustworthy nursemaid, Impa, to secretly escape into the land and go find a man with enough courage to destroy the evil Gannon. Upon hearing this, Ganon grew angry, imprisoned the princess, and sent out a party in search of Impa.
Braving forests and mountains, Impa fled for her life from her pursuers. As she reached the very limit of her energy she found herself surrounded by Gannon’s evil henchmen. Cornered! What could she do? … But wait!
All was not lost. A young lad appeared. He skillfully drove off Ganon’s henchmen, and saved Impa from a fate worse than death. His name was Link. During his travels he had come across Impa and Ganon’s henchmen. Impa told Link the whole story of the princess Zelda and the evil Ganon. Burning with a sense of justice, Link resolved to save Zelda, but Ganon was a powerful opponent. He held the Triforce of Power. And so, in order to fight off Ganon, Link had to bring the scattered eight fragments of the Triforce of Wisdom together to rebuild the mystical Triangle.
If he couldn't do this, there would be no chance Link could fight his way into Death Mountain where Ganon lived. Can Link really destroy Ganon and save the Princess Zelda? Only your skill can answer that question. Good luck. Use the Triforce wisely.
(Text found from Zelda.Wikia.com: The Legend of Zelda: Prologue.)
That story came from a couple of pages from the manual, where this story appeared on top of a drawing of the land of Hyrule:
Edit: I decided that people may enjoy me providing a convenient hyperlink to the actual introduction sequence in action, where the summarized text is shown in the actual cartridge. If you'd like to gaze upon this, see: 15 seconds into The Legend of Zelda Intro (video on Youtube)