The narrator is—sorry to spoil it for you—pretty much Kurt Vonnegut. (Not that that's a bad thing: Vonnegut, as you can tell from reading this book, is awesome.) He talks from the first person perspective about the experience of writing this very novel, and he mentions lots of details about his own life that match Vonnegut's own biography.
So why don't we just call him Kurt Vonnegut? Well, first off, he doesn't call himself that in the book. And second, even if this guy is a stand-in for the real-life Kurt Vonnegut, he is still a character within a fictional novel. There are lots of real details in the narrator's account of himself... but he is also a fictional device meant to tell a story.
The narrator is caught up in this contradiction: he was at the firebombing at Dresden (see "In a Nutshell"), he saw it, and yet, because it is so personally enormous to him, he can't find the words to write about it.
The narrator compares himself to Lot's wife, the Biblical figure who was punished by God for turning back to look at the destruction of Sodom and Gomorrah in Genesis 19 (she was turned into a pillar of salt). Although he knows about the terrible things Germany did as a nation during World War II—a man at a party, for instance, tells him that the Germans "made soap and candles out of the fat of dead Jews" (1.6.6)—he still can't help but feel pity and sorrow for the thousands of ordinary Germans killed in front of him during the war.
He can't help himself. He has to look back at the destruction of Dresden, even though the citizens of Dresden—like the poor suckers in Sodom and Gomorrah—were dismissed as being sinners. Because he spends the book strolling down a pretty unpleasant stretch of memory lane, he thinks that it means that his book will be "a pillar of salt" (1.22.3).
The narrator keeps popping up throughout the "fictional" part of the novel revolving around Billy Pilgrim. When Billy first meets Wild Bob, the narrator comments that he and his war buddy, Bernard V. O'Hare, have both been to Wild Bob's hometown of Cody, Wyoming. When Billy stumbles into a latrine in the British POW compound in Chapter 5, one of the American soldiers with explosive diarrhea (ew) is the narrator himself. And when Billy arrives in Dresden and one of the other POWs comments that the city looks like Oz, that's also the narrator.
We don't think there is one single answer for why the narrator keeps cropping up throughout the book as one of fictional Billy's fellow prisoners. Maybe he's just trying to keep the reader's interest in Billy's adventures?
We think it might also be that Billy is a really extreme depiction of a clown in the middle of a war beyond his control. Some readers might start getting angry or offended at such an unflattering, unheroic depiction of the American soldier. (For more on this point, check out Mary O'Hare's "Character" page.)
But the narrator himself really was in the war and really did get explosive diarrhea and really did think that Dresden was a lovely city (a lot like Oz). His experience of war wasn't romantic or heroic... which is exactly what gives even this fictional novel its sense of sadness and futility.