Slang for alcohol. Drinking enough of any type makes you move like a Muppet where your arms and legs move oddly and you stare at people strangely while talking to them.

Muppet juice is another word for meth. When meth addicts lose their teeth( and most do) they look like Muppets (puppets made famous by Jim Henson. Such as the Muppets and fragil rock)

Meth being the "juice" that causes this. "Muppet effect"