The task begins in Kakariko Village. Talk to Lady Impa, who will provide further clues. In truth, the entire mission revolves around Impa granting you more quests: Locked Mementos and Free the Divine Beasts.
Reach the Lab in Hateno Village (1). Talk to Purah, and then to Symin, and then go back to Purah again. She will send you to get the blue flame needed to light the furnace in the lab. Before you set out, be sure to take a torch with you.
It is best to go to the location of the flame on a sunny day. When it rains, the torch you lit will simply go out. It will also go out if you put it away, run or climb. Take the flame back to the lab. Along the way, it is worth to light stone torches, because when yours goes out for some reason, you will not have to go back to the very beginning. When you are back use the flame to light the furnace.
Go inside and talk to Purah. Sheikah Slate will gain new features, e.g. a camera. Use it to take a photo of Purah, and then show it to her. Afterwards, return to Kakariko Village and talk to Lady Impa.