There were a couple a version but in sum:
Ucanogos was a beautiful daughter of an Indian chief. She and Timpanac, a strong brave from another tribe, fell in love. The chief came up with a contest(although in one version the daughter came up with the contest so the chief did not make her marry the richest man) and whoever won would have his daughter's hand in marriage. The challenges included killing a animal with bare hands, racing around the lake, and climbing to the top of the mountain. Timpanac was the bravest and strongest and was sure to win and the other braves were jealous. They murdered him by throwing up off the mountain. Ucanogos was so heartbroken by his death that she died of a broken heart in her fathers arms (another version was she threw herself off the same mountain).
The heart of the Timpanogos caves is said to be the symbol that their spirits embraced and they are united in death.
For anyone interested, I love this legend. Nature is my greatest source of inspiration where writing is concerned, and I spent a lot of time gazing at and hiking around Mount Timpanogos, as well as a trip into the Timpanogos caves to get ideas for Cobbogoth. I've lived with Timanogos at my back for the last ten years, and I think I'd feel very weird without it there. Enjoy the sights!