Why does Seymour commit suicide? This is possibly one of the most highly-debated short story questions of the last fifty years. There are dozens of theories, and we can't be sure which one of them is "right." It could be that Salinger wrote his story with some specific reason in mind; or it could be that he intentionally left it ambiguous. It also might be, as we argue in "What's Up with the Epigraph," that the "answer" to this question can't be logically conceptualized. In any case, here's a little pupu platter of Seymour theories:
#1) Innocence, Children, and the War
Let's not forget that Seymour's mental troubles are the result of the war, and that he's suffering from what today we would probably call post-traumatic stress disorder (though this term wasn't yet around when Salinger was writing). We infer that Seymour has witnessed some awful things during his time in the service, and that he's having a hard time readjusting to being home. We see that he's retreated into a largely insular world, and that he's no longer comfortable interacting with most adults. Sybil offers him a glimpse of the world as he would like it to be – innocent, curious, and pure – but his interaction with the woman in the elevator reminds him that the adult world is actually nothing like this. Unable to cope with reality, and unable to function normally, Seymour turns to suicide.
#2) Seymour Is Enlightened
This theory makes more sense if you've read some of Salinger's other works about Seymour, or in particular if you're looking at "Bananafish" as part of the collection Nine Stories. As hinted at in the epigraph to Nine Stories, there is a common theme of Zen Buddhism in Salinger's work. "Bananafish," the first story of the collection, and "Teddy," the final story, both deal with this theme, though the latter far more explicitly. In "Teddy," a young child genius is somewhat of a Zen master. He discusses his flirtation with enlightenment in a previous life, and he casually foretells his own death.
It's interesting to note that both "Bananafish" and "Teddy" end with the death of the main character. At first, the tone of these deaths may seem very different. Teddy calmly accepts his accidental death as a step on the road to enlightenment, and there is tranquility even in the jarring ending. What of the conclusion to Bananafish, though? Is it a jarring, painful ending, quite different from that of Teddy? Or is Seymour's death, too, a calm and accepting step in the right spiritual direction? In "Teddy," for example, the title character explains that death is in many ways like waking up. It's no coincidence that Muriel is sleeping in the bed nearby when Seymour puts the gun to his head. He's waking up; she's still asleep.
Consider the idea of the bananafish. We flesh out this idea a but more in "What's Up with the Title?" The short version, though, is that the gluttonous bananafish may represent the material obsessions of people. Seymour doesn't want to be like the bananafish, pigging out on physical desires, so he kills himself. He ends his physical existence, but not, many argue, his spiritual one.
#3) Seymour is Sexually Attracted to Sybil
One possible, if far less satisfying, reason for Seymour's suicide is pedophilia. He's attracted to Sybil and even goes so far as to kiss her foot. He's then filled with shame at his action and so kills himself, preserving Sybil's purity in the process. It's unlikely that Salinger intended this as a line of reasoning, but there you have it.