It refers to something being meaningless, excessive, and non of one’s business.
Moofy is an animal with thick, fluffy fur, and therefore it doesn’t need a blanket.
However, it may not actually be “unnecessary care” for Moofy, since it does like to hide and huddle under blankets in the middle of winter. It is also a proverb that often receives witty remarks, such as, “the behavior would be unnecessary even for other creatures, if it was summer…”
Translation note
Moofy: A small herbivore, also called furballs because their entire body is covered with fluffy fur. They so docile that they are thought to have no aggression whatsoever. When they are caught by a predator and are about to be swallowed, they instantly swell their tail to several times its size and make the predator choke so that they can be spat out. This unique and rather peaceful defense style has led to many proverbs and sayings related to this behavior. Although they are honored for the usefulness of their fur, because they seem as if they do not care about anything (especially in captivity), they often appear in proverbs and folklore as an expression representing indifference, both good and bad.
Reference
“Dictionary of Proverbs by Biological Species ‘Moofy’” (Direlle Aarti)
“Dictionary of Global Proverbs ~with Detailed Explanations of Quips” (Reeda Supianne)