Despite all the turmoil and uncertainty of the country they lived in, the Potters managed to keep the existence of their daughter a secret, not only from their enemies and the 'media' of the Wizarding World but also the allies that they didn't necessarily fully trust. Someone amongst the allies who were kept at an arm's length was the Headmaster of Hogwarts, Albus Dumbledore. While the man was a figure of great influence, the Potters doubted that he wouldn't do something stupid if he was in the know regarding Fauna's existence and the young girl's… Eldritch parentage… even if she definitely looked more human than her Eldritch parent.
Something that would prove itself to be a mistake on the Potters' part was them putting their trust in a person who they believed was a friend. Peter Pettigrew. Also known as Wormtail. A name that proved itself to be an appropriate one when he sold out his friends out of fear. Realizing too late that they've been betrayed, the Potters also realized that they wouldn't be able to remain in the life of their daughter for much longer, so they called upon the Moon Presence in order to let the Great One uphold its end of the bargain and place its child in the care of its agents.
The night when the Moon Presence came to collect its child coincided with the night when the human calling himself Lord Voldemort invaded the home of the Potters and proceeded to kill them. In fact, if the man hadn't screamed in understandable terror at the sight of the Moon Presence reaching into the room that was clearly the room of a relatively newly born child through a window and picked up the surprisingly calm child from its creedle (not that Lord Voldemort was likely to ever admit to having screamed in terror), the Great One would most likely not have noticed his presence. Not that his presence did much as the Great One simply continued with what it had come to do with complete indifference.
In a vain attempt to get the Great One's attention, Lord Voldemort shot a rather nasty curse in its direction, only for one of the many tentacles sprouting from the Great One's head to absentmindedly slap the curse away from it and the baby in its hands as the Moon Presence moved its body out through the window and starting to ascend towards its domain. However, before it made its exit from the Waking World, the Moon Presence made sure to get rid of the puny human who was making a fuss for its attention by dragging it along by the ankle until it reached a sufficient level of altitude and unceremoniously dropping the human, leaving nothing but a messy stain on the ground as evidence that the human had been there.