Why do Jewish boys turn to men at 13?

Answer 1

Every system of laws incorporates the concept that there is an age at which a person attains
the maturity to be held responsible for certain adult duties that were denied to the child.

Example:

In the USA, a person under the age of 18 is considered insufficiently mature to purchase beer, or to vote
for candidates for elected office. A person under the age of 21 is considered insufficiently mature to sign
business contracts.

In Jewish law, a male is considered sufficiently mature from the age of 13, and a female from the age of 12,
to be held responsible for the standards of conduct and free choice required of an individual by the ethics
and standards of Judaism.

And that's what "bar mitzvah" and "bat mitzvah" are all about.

The main reason why these occasions are seen as occasions for celebration by parents is that before
the child reaches that age, guess who is responsible and accountable for the child's misbehavior . . .

Answer 2

The Talmud says that a boy becomes an adult at puberty. In some parts of the text, this is defined biologically, for example, in terms of growing pubic hair, but in other places, age 13 is specified. That is a fair approximation of the age of puberty, and using a simple age avoids having to conduct a potentially embarrassing physical exam to see if a boy is biologically, and therefore legally an adult under Jewish law.