In today’s day and age, it is difficult to fathom how the rabbis could deem a twelve or thirteen year old child even remotely mature enough to be considered an adult. In truth, however, reaching the age of Bar or Bat Mitzvah has never meant that one was a full-fledged grown-up, but simply that one was now responsible for his/her own actions. Becoming "responsible" is an important part of human development.
In laying out the course of a man’s life, particularly in relation to study, Rabbi Judah ben Tema said (Ethics of the Fathers 5:25): "A 5 year old begins [learning] Scriptures, a 10 year old begins Mishna, a 13 year old becomes obliged to observe the commandments, an 18 year old goes to the marriage canopy, a 20 year old begins earning a livelihood, a 30 year old attains full strength, a 40 year old attains understanding, a 50 year old can offer counsel..." Rabbi Judah’s statement continues until 100 years old, noting the characteristics of different ages.
As Rabbi Judah ben Tema pointed out, there is an appropriate order of development. A boy/girl must learn to be responsible for his/her own actions in his/her parents’ home before becoming fully independent (around age 18 or 20). This process begins, in earnest, at the age of Bar and Bat Mitzvah.
Perhaps the classic celebration speech should more correctly read: Today, I am learning to be a man/woman.