The Haney Egg Layin' Inducin' Machine has Vera, the hen, lay an egg. It goes down a series of chutes, where it's candled before triggering a bell. That cues Raul, the rooster, to pull a string that feeds Vera. This induces her to lay another one.
The sign on Mr. Haney's truck announces "Haney Chicken Raising Equipment Company, (member of the Fryers Club)."
The title is a line from the childhood song "Old MacDonald Had a Farm" which came from several sources and was used as a memory exercise by remembering and repeating all the previous verses (animals).
The play on words that Mr. Haney confesses to when Mr. Douglas notes that the sign on the truck says, "Fryers Club," has to do with the Friars Club in New York City, founded in 1904 at the behest of Broadway producers and publicists. It is famous for its comedy roasts and the stellar list of past and present members, which includes George M. Cohan, Will Rogers, Irving Berlin, Lucille Ball, and Barbra Streisand.
The slide rule that Alf and Ralf use is a simple mechanical calculator comprised of parallel rulers with numbers spaced in such a way that multiplication, division, logarithms, and other such calculations may be made by sliding the rulers one way or the other in order to arrive at a correct answer. This devise has been in use since at least as far back as 1622 and was still in great demand when this episode originally aired in 1969. With the coming of electronic calculators in the 1960s, the use of the slide rule has greatly declined and, for certain applications, entirely disappeared.