Although we never learn the narrator's real name, he's still central to the novel, more for what he tells us about Holly than for what he tells us about himself. In fact, we actually learn very little about him throughout the story. We know that he's a writer, that he has moved to New York to pursue his career writing short stories, that he finds Holly's life very strange and different from his own, and that he eventually falls in love with her (although the nature of this love is never fully defined). But beyond that, he remains kind of a mystery to us.
We do get glimpses of his personality when he disagrees with Holly, when he feels hurt by the things she says and does, and when he experiences great sadness at the thought of losing her. However, this all comes to us in short bursts and we never really get inside the narrator's head. Why does he let her into his bedroom in the first place? What did he do before becoming a writer? What is his family like? We never get the answers to these types of questions, or even to more mundane ones. For example, when he tells us about his new job he offers little in the way of actual information:
"Because toward the end of the month I found a job: what is there to add? The less the better, except to say that it was necessary and lasted from nine to five." (8.1)
Who knows what he does all day every day? Not us, since he doesn't tell us.
The narrator functions more as a way for us to learn about Holly. It's through his eyes that we see her, through his descriptions that we get to know what happens to her throughout the course of the novel, and it's in his words that we read about her. In this way, we might consider the narrator less as a fully-developed character and more as an important literary device. He paints the picture of Holly Golightly for us, and without him we'd really have no story to read.