In Slaughterhouse-Five, so many of the incidents the narrator describes are so far-fetched and unbelievable that it's almost necessary that Vonnegut use a first-person narrator. Having Billy Pilgrim talk directly to the reader allows the reader to empathize with him because we hear the story from his voice, and he even admits his sense of disorientation when he time travels. Billy describes both his war experiences and his time on Tralfamadore, where he is supposedly being studied by aliens.
The novel offers an interesting take on PTSD, which Billy is clearly suffering from as a result of his war experiences. Billy tells the story as though all of these surreal events are actually happening to him, but on a more allegorical level, we could look at his time travel and experiences on Tralfamadore as symbolic of the scattered state of his mind after the war. In being "shell-shocked," his mind is fragmented, and events he's experienced feel surreal because they are so traumatic. Having Billy tell us about his travels makes it easier to follow his logic and feel sympathy for him. If we were to view Billy's story from an outside perspective, we might think he were simply insane. Vonnegut's narrative strategy allows the reader to understand Billy's thinking and to access the allegory.