The Muddy Key is in the path of the Fields leading to the Hills. It is near the small rice field where the Woman in the Rice Fields first appears to attack the Protagonist upon picking up her Broken Necklace. You'll need to get around the Spirit in order to pick up the key.
The Muddy Key is used to open the door of the steel fence that divides the Fields from the Hills. It is needed to progress the story, as the Protagonist needs to reach the Spirit's dead body at the bottom of the Cliff.