To being this quest, head to Oasis at the north of the map. Inside you will have to undergo a ritual to be allowed to speak with "Him." Once you do so, you'll be able to speak with Harold (the only person to appear in all three Fallout games) who has essentially been overtaken by the roots that were growing on his head, become rooted in this spot. He asks you to end his suffering and destroy his heart. He also advises against burning him because it would hurt.
Leave his chamber and head toward the map marker to find two of the Oasis residents talking. After their conversation, Birch will give you some sap and Laurel some linment. Continue on to the caverns and head to Harold's heart. You now have a few options: destroy the heart and kill him, give him the sap to stop his growth or apply the linment to increase his growth. Whichever you choose, head back to Harold to finish the quest.
See Oasis for more specific info.