Why does Billy time travel and what is its significance in "Slaughterhouse-Five"?
In the book, one major life-changing event appears to be the catalyst for Billy's non-linear time travels: his war-induced PTSD (Post Traumatic Stress Disorder).
We know that Billy's devastating experiences during and after the bombing of Dresden severely traumatized him. Even after he returns from the war, he experiences extreme fluctuations in emotion; he is often anxious and exhibits symptoms of hyper-arousal, which are commonly associated with PTSD sufferers. Billy's conception of time and place continually evolves, and his time travels appear to function as a clever metaphor for his dissociative disorder.
Like all PTSD sufferers, Billy often finds himself emotionally incapacitated by "triggers" that force him to relive his war-time experiences during the most inopportune moments. For example, during his and Valencia's anniversary party, Billy becomes discomposed when the barbershop quartet of optometrists sing "That Old Gang of Mine." The song triggers Billy, and he becomes embarrassed by his reaction to it. He imagines that he has a "great big secret somewhere inside," but he is afraid to explore it.
Billy copes by repressing much of his emotional angst and compartmentalizing his life into unrelated segments of time: he alternates between his POW experiences in Luxembourg, his married life to Valencia, and his bizarre exploits with the Tralfamadorians in their immortal world. In a very real sense, the Tralfamadorian concept of simultaneous time is an appropriate metaphor for Billy's severe PTSD. Billy can make little sense of his life because his war experiences have severely curtailed his ability to be objective.
He also lives a surrealistic existence because of his flashbacks. During these times, we are forced to question where Billy begins and where the war ends. The sad answer is that the war doesn't end for Billy; it's always with him. In the book, the Tralfamadorians explain how they approach life: they "spend eternity looking at pleasant moments." Since evil coexists on multiple plains with goodness, Tralfamadorians "ignore the awful times, and concentrate on the good ones."
So, Billy's time travel is not just significant as a metaphor for his dissociative PTSD, it also highlights the necessity of a positive attitude in the journey towards wholeness.
In Vonnegut's Slaughterhouse-Five, Billy's time traveling is his experiencing what all Tralfamadorians experience. The aliens experience all of existence at any given time. Thus, they see their existence as a whole. They see consequences and repercussions of their actions at the time they act.
That's the point of Billy's time travel. Humans don't see the whole picture. Humans don't see the consequences and repercussions of their actions. The implication is that, if humans could see, or would think about, the whole picture, they wouldn't do things like bomb the city of Dresden, treat each other cruelly, etc.
Vonnegut uses the Tralfamadorian view of their lives, and Billy's time travelling, to demonstrate this point. Vonnegut was an atheist, but he was not a nihilist. He satirizes humans with the hope that they will learn to behave better.
At the same time, I would be amiss if I didn't point out another side of the issue. Since neither the Tralfamadorians or Billy can change the future even though they see it, being "unstuck" in time suggests a lack of free will. Billy sees his death before it occurs, but is unable to or at least does not stop it. And the aliens know that they will cause the destruction of the universe, but will do it anyway.