Question

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

Answer 1

I believe that the movement and faith you're referring to here is Christianity.

One of the major beliefs in Judaism is that Messiah will come: and the new movement believes that the Messiah has come, and that this is Jesus Christ.

The old belief was not to be reconciled with the new one (how can one wait for the Messiah if he has come already?) and the new movement had to split out: today it's the religion, Christianity.