One of the risks in a no-narration "Double Indemnity" would have been failure to empathise with Neff. After all, it's pretty nasty stuff he gets involved in. But most of us know the intoxication which can be visited by the opposite sex, and hearing this vividly through Neff's narration allows us to stay with him as he slides. Off topic: what a lovely exchange to end the movie, between two hard-nosed insurance men, as Neff says to Keyes "Guy you were looking for was too close. Right across the desk." Keyes replies, uncharacteristically and movingly, "Closer than that, Walter". Neff replies "I love you too".

If you're a fan of the movie, you probably know that Billy Wilder actually conceived a different ending where Neff goes to the gas chamber. You can read about it here and hear Wilder discussing it here.