After the Professor invents a mind swapping device, The Professor and Amy decide to swap bodies so that the Professor can be young again and Amy can eat whatever she wants.
They soon learn that you can't swap back, so Bender agrees to be a third body, but just steals Amy's. He tries to use her body to help him pull off a heist to steal the Robo Emperor's wealth. After he fails, he ends up swapping bodies with the Robo Emperor and a wash bucket, which he claims to be his bending body, so the emperor can feel poor again.
The wash bucket swaps with Amy, who goes to confess its love for Scruffy, who rejects it. After all, in our society no one understands Janitor-Wash Bucket love.
Fry and Leela swap their bodies with Zoidberg and the Professor respectively and attempt to prove they love each other for their inner beauty, not outter, and get it on. Gross.
Amy, meanwhile, ends up in Leela's body and turns it fat, so Hermes agrees to swap and they both end up losing their weight back to normal after Hermes uses disclipine and Amy loses her apetite after seeing Fry and Leela make out as Zoidberg / Professor.
In the end, everyone is able to get back to their bodies thanks to some math by the Harlem Globtrotters.