There comes a time in every story (hopefully) when the words fly from your fingertips onto the page in a voice that you come to realize is not your own. And it shouldn't be -- unless you're writing an autobiography of course and then the voice not being yours is just weird.
The voice that you find in the story isn't yours because it's the voice of your character. This is especially true in first-person stories where you're sitting directly inside the head of the protagonist (or antagonist as the case may be) and seeing everything they see, but it's also valid in third-person limited and third-person omniscient tales.
When a writer says a story "wrote itself," then, what they mean is the character's voice came through loud and clear and fully formed without an ounce of hesitation. Moriah's sarcasm and deep-seated anger were the first traits I discovered without hesitation when I started writing CALLARION AT NIGHT; her feelings of inadequacy and burning desire to find the truth came later. And there are times when her voice comes through more clearly than others too -- like when her father dies or during The Argument with Nicolai that shifts them from dancing around the shared pain that keeps them apart.
What about you? When do you hear your character's voice the clearest?