The whole article is a little hyperbolic, but the distinction is important. To "lie" is to position one's own self, while to "lay" is to position some grammatical object. Confusingly, "I will lay myself down" is grammatically correct, because "myself" serves as the object.

There are situations in which using the wrong verb conveys a different meaning. "Please lie down on the floor" is a request to put yourself horizontally on the floor, while "please lay down on the floor" is a request to spread soft feathers underfoot.

I'm sure it's a little bit tongue-in-cheek.