Since "re" means "again," to be reborn means to be "born again." Since no one really has the chance to experience their birth a second time around, to be reborn means to undergo a meaningful spiritual change.
In certain Fundamentalist and Evangelical Christian churches, congregants choose a time to be baptized, or ritually admitted to the church, and it's at that moment they consider themselves to be reborn. This idea is rooted in a section of the Bible's New Testament, commonly translated so that Jesus says, "no one can see the kingdom of God unless they are born again."