The Catcher in the Rye uses first person narration because this is the most direct access to Holden's thoughts. This novel is about Holden "coming of age" or, rather, it is about his reluctance to grow up. As Holden narrates, the reader gets Holden's perspective of the world. His thoughts, his inner monologue, and his expressions are presented with some degree of stream of consciousness which demonstrates a freer, less rigid, or less structured translation of thought. This is symbolic of Holden's antipathy towards social conventions and structured ways of behaving and thinking.
The language he uses to narrate is often childish. This is a manifestation of his unwillingness to speak or even think like an adult because he thinks that adult world is phony. So, not only does Holden try to avoid acting like a phony, he also tries to avoid speaking and thinking like one. Using first person narration is the technique that allowed Salinger to show this.
First-person POV was essential because it gives the reader full access to the narrator's mind, where the crisis was taking place. In third-person, the reader would have been constantly distracted with comments such as: "he was wondering" or "he thought to himself." Second-person would have worked but it's a tricky point of view that would trip up most readers.
First-person also has a natural flow and feel of story-telling. It's the voice that most people use in oral story-telling. Stand-up comedians also use first-person point of view because it's so natural and effective. The narrator has complete control over how much or how little of his or her thoughts to reveal, but the reader/listener feels as though they're being confided in. Consequently, there's a feel of authenticity and intimacy than with third-person.