One of Billy Pilgrim's big obsessions is this unchanging idea of time that he claims to get from the toilet plunger-shaped aliens of the planet of Tralfamadore. (Seriously.) According to the Tralfamadorians, each single moment in time goes on forever. It is an Earthling illusion to think of time moving forward or backward, because all of these moments take place at once, simultaneously.
This matters because Billy's ideas also influence the shape of the novel itself. The weird thing about the ending of Slaughterhouse-Five is that it actually comes in the beginning and the middle of the novel as much as it does at the end. In the final section of the book, we get scenes of: (a) the aftermath of the Dresden firebombing (see "In a Nutshell" for more on this); (b) the execution of high school teacher and American POW Edgar Derby for stealing a teapot from the ruins of the city; and (c) a bird tweeting at Billy Pilgrim.
All of that stuff, though – the firebombing of Dresden, Edgar Derby, even the bird – also comes into the first chapter of the novel. In the beginning of Slaughterhouse-Five, the narrator tells us that this novel is about Dresden, he discusses poor old Edgar Derby with his war buddy, Bernard O'Hare – hell, he even informs us that the novel "ends like this: Poo-tee-weet?" (1.22.12).
In other words, throughout the book, references to future and past events intermingle. For example, we know that poor old Edgar Derby, even while he's walking around in Billy's memories, has already been shot. At the same time that Billy listens to a barbershop quartet at his wedding anniversary party in Chapter 8, he remembers his German guards seeing the destruction of Dresden in 1945 (8.22.1). There is a real sense that everything happens simultaneously in this book, and we just get to some things before others thanks to the order of the pages.
One last word on the last word of the book: right in the first chapter, the narrator tells us that birds say "[a]ll there is to say about a massacre, things like Poo-tee-weet?" (1.15.2). And the final phrase of Slaughterhouse-Five, following a depiction of the massacre at Dresden, is, of course: Poo-tee-weet? This book is like one of those nested Russian dolls where you open one and inside is a smaller version of the exact same thing: nearly every chapter foreshadows what's coming next and looks back at what has just come before it. So it's hard to say where the stories of Billy Pilgrim and the narrator truly begin and end, except that there is a first page and a last page to the book.