"The Glow-Worm" is a song from Paul Lincke's operetta Lysistrata. It was written in 1902 with German lyrics by Heinz Bolten-Backers as "Das Glühwürmchen". The song was translated into English by Lilla Cayley Robinson and used in the 1907 Broadway musical The Girl Behind the Counter. A later version by The Mills Brothers, with expanded lyrics by Johnny Mercer, became a hit in 1952.
The instrumental version of the song, in the form of a gavotte, has been used in a sketch repeated by the Muppets on several occasions. It's usually referred to as "Glow Worm" or "Inchworm" as a reference to one of the characters.
The sketch involved Kermit humming the tune while sitting on a wall as a worm crawls to him and taps him. Kermit sees the worm and eats it, followed by another worm who shows up and does the same thing. The third time a worm shows up, Kermit catches it early, but it gets longer and longer until it's revealed to be the nose of a larger entity (often Big V), who eats him.
According to the book Jim Henson: The Works and Jim Henson's Red Book, the sketch originated on Sam and Friends. In this early incarnation, Kermit is bothered by two fingers, which he eats. When a third one appears, Kermit pulls it up, revealing an entire human hand. The two inspect each other before the hand tickles Kermit off the wall.
The animated Fraggle Rock series paid tribute to this sketch. In a few episodes, a Fraggle bug collector shows up in the background, picking up little worms from the ground. The worm turns out to be the trunk of a large monster.