How did a new movement within judaism lead to the development of a new faith?

I think you refer here to Christianity as the new faith.

Christianity essentially accepts all of Judaim, but it adds the teachings of Jesus Christ on "top" of the teachings of Judaism. Judaism already had a belief in a Messiah to come and Christianity believes that this was the Messiah. So the Jews that accepted Jesus as the Messiah were no longer Jews, as Jews still wait for the Messiah.