Why do we have "handedness"?

Why are people either right-handed or left-handed? What possible benefit does that have over being ambidextrous? I find I am right-handed, right-footed, and even right-eyed! When I wore single-muff headsets on my job, though, I preferred them on my left ear and not my right, so that's a bit of a question.

And, do animals also display handedness?

Thank you.
Ken Silva
Laveen, AZ
USA