Tim O'Brien is a master of storytelling. In fact, he's a master of storytelling about storytelling. It's called metafiction. Fiction about fiction. This is a story about a story, or storytelling.
When someone is about to tell a tall-tale, a storyteller will preface it with "This is one I've never told before. Not to anyone" or "You won't believe this story; you can't make this stuff up..." or some such jazz.
It's a way to draw the audience in to his supposed confessional. We know it's fiction, but with that preface it psychologically blurs the lines a bit, and we ask ourselves: is this true?
"On the Rainy River" is, ironically, the most fictional story in the novel of memoir-esque stories. There was no trip to the border, no Elroy. It's all metaphorical. Tim went to war, if not directly, then soon enough. He didn't run away. So, why would he make it up? He wants story truth to be more true than happening truth. It's always been that way, right? No one wants a rough draft of the truth instead of a well-crafted tall tale.