Vulgar Etymology Contest

Keach Hagey reports:

Rachel Donadio reported on that story for the New York Times yesterday in a piece that, while quite colorful, was still missing a few hues.

As Donadio told the Italy’s La Repubblica, the Times editors, citing the paper’s rules against vulgarity, forced her to cut out the wiretaps’ irresistible quote: “She was his little girl, I’m his piece of ass.”

I have an embarrassing confession -- I don't really understand that phrase. I know what it means, I just don't understand where it comes from, or why it means what it means. Especially the "piece" part.

Commenters, please explain this one for me as cleanly as possible. I will reprint the explanation that best combines the traits of being convincing and non-flithy.