Influences are among the most mysterious objects in the Big-Big-Picture. There is one for every world in the universe. They seem to serve as a kind of celestial counterpart for a world's Parental Totality.
They are called factories by the Goofy Service Jerks. Pod'l'pool world influences are housed in a Service Station which is manned by Goofy Service Jerks. Before the population of a Pod'l'pool world, such as the Beanworld, can increase, its Parental Totality must send out a midnight signal. The signal is caught by the Boomrail that makes up the walls of the Service Station and transmitted to the Influence inside. The Influence turns over and out pours a supply of Reproductive Propellant, which is caught by waiting Goofy Service Jerks who then deliver it to the Parental Totality of the calling world. The Jerks, who ordinarily have little direction sense, are filled with purpose by the Repro, which guides them to the proper destination. Shortly after the Repro is consumed by it, it grows a Pod'l'pool which serves as a combination womb and nursery for the new infants.
Mr. Teach'm tells us it is said that worlds and Influences used to live together, but something happened to scatter the Influences all over the Big-Big-Picture. Heyoka, after breaking out, immediately floated away into the sky, eventually coming to the center of the Beanworld Influence, which turned over and changed shape. The Jerks said she "altered the specifications of the factory," but it is uncertain what that means.
Spoilers follow....
In "Remember Here When You Are There!", Heyoka finally leaves the Influence by swimming through the air. If she stops swimming, she is drawn back to the Influence. This may indicate something about the nature of gravity in the Big-Big-Picture, considering that Heyoka does things in reverse.