There is a very long and detailed history of Reiki that I invite you to read here. Following is a brief, condensed version of how the healing art of Reiki came to be.
Dr. Mikao Usui is often credited as the founder of Reiki. However, Reiki is thought to have been around for thousands of years prior. The legend of Dr. Usui goes as follows. Growing up a Zen Buddhist, Mikao became fascinated by western culture and began studying allopathic medicine. He became very ill and had a spiritual awakening. He joined a zen monastery and “discovered” a method of healing along with specific hand positions. The legend explains that he wanted to have more awareness of this practice and so he went up to Mt. Kurama on a 21 day fast. He meditated during this time. There was a bright light that flashed above him, going directly into his forehead, showing him the symbols. He began his journey down the mountain and injured his ankle. He put his hands on his injury and noticed the healing that began. Along the way back down the mountain and back into the monastery, Usui was able to heal several other people with his hands. After healing some people who ultimately returned to their old habits, Usui was disheartened. He went into a period of meditation again and was given the 5 Reiki principles. This was when he was reminded that it is essential to heal the body, mind and spirit together.
That is the legend. Again, you can read more in depth details of the legend and the life of Dr. Usui here.