What happens in bar mitzvah and how this show that the child has joined the religious group?

To become a bar mitzvah is to become a man in Judaism. There is an event, when the boy is thirteen, where he reads from the Torah for the first time. After reading from the Torah, he is a man, and must take full responsibility of his actions (meaning wearing a tallit and tefillin, not sitting next to women in synagogue, etc..)