I've always interpreted Supper's Ready as the story of the end of the world. Each section deals with different people and how they're dealing with their fate. The beginning of the song (Lovers Leap) shows a couple living their last few days together and trying to stay happy. While they're physical together they're minds are elsewhere most likely thinking of their eventual deaths.
The next section shows people who don't think anythings going to happen. They go on with the rest of their lives. The farmer continues to care for his farm and the fireman still puts out fires.
Ikhnaton And Tuscon And Their Band Of Merry Men and How Dare I Be So Beautiful tell that some people, fearing they will not make it into the next world (heaven), begin to fight to see who's superior.
The following two sections (Willow Farm and ALL CHANGE) discuss the people who can't handle the Apocalypse and go insane. Claiming they can change their form and that they are self-sufficient their delusions become more and more intense until they become completely insane.
The rest of the song describes the actual end of the world. Trumpets proclaim that "the time has come" and the world quickly falls apart. We finally see who (if anyone) chose the right way to spend their remaining days. The two lovers have made the right choice by enjoying their last hours. The song finally ends when everyone is taken to the new Jerusalem.