As stated in the Wiktionary page, this expression is referring to meaningless puns made by appending an unrelated sentence starting with the last syllable(s) of an existing one. The figure of speech is named Dorica castra.
The first and best know one is:
Comment vas-tu ? (How are you ?)
Comment vas-tu yau de poêle → Tuyau de poêle (Stove pipe)
Pas mal, et toi ? (Not bad, what about you ?)
Pas mal, et toi le à matelas → Toile à matelas (Mattress ticking)
from Henri Maigrot published in the 1896 Almanach Vermot.
The nursery rhyme Trois petits chats is using a similar approach:
Trois petits chats, trois petits chats, trois petits chats, chats, chats
Chapeau de paille, ...
and so on
Translations keeping the joke spirit are left as an exercice for the reader ...