Does anybody know why a boy is obligated to keep mitzvos after he gains sexual maturity (13 years old)? How is the possibility of fathering children connected to that?
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Let's start with the fact that the usual description is that a child below this age doesn't have daas.
What is daas? R' Shalom Dovber Schneersohn zt"l, the fifth Lubavitcher Rebbe, defines it (in Kuntres Hatefillah ch. 5) as the ability to empathize. A young boy or girl may be smart enough to understand something intellectually (such as that being poor is hard), but he or she is not capable of feeling it (say, for example, being able to identify with a poor person).
Daas, as so defined, is critical to accepting one's responsibilities and privileges as a Jew. (Reshimos of the Lubavitcher Rebbe zt"l, no. 19)
Going further, Kabbalistic and Chassidic literature see the two sefiros of chochmah and binah (loosely, "wisdom" and "understanding") as the spiritual analogues - actually, as the spiritual sources - of male and female. Their union, which engenders children (in terms of the sefiros, these are the emotional attributes - chessed ("kindness"), gevurah ("severity"), and so forth), is daas, which is why the marital act is called "daas" in the Torah, as in Gen. 4:1. (R' Shneur Zalman of Liadi, in Tanya ch. 3, points out that indeed without daas it is impossible to develop true love and fear of Hashem.)
Thus, a boy below 13 years old, and a girl below 12, lack mental daas - and therefore also its physical analogue, the ability to have children. The definition of "maturity," then, is the age at which they gain both of these kinds of daas.
Rashi on Avot 5:21 brings an opinion that it is Halacha L'Moshe M'Sinai (Law Given to Moshe on Sinai) that once a male reaches puberty (2 pubic hairs) he is considered an adult and obligated in Mitzvos.
The Rabbis figured out that the average male reaches puberty at 13 and therefore established 13 as the default bar mitzvah age.
Halacha L'Moshe M'sinai is usually associated with the idea that the Halacha was given without a reason.
Even more (assuming I understood it correctly), it appears that the Maharil (Shu"t Maharil paragraph 51) says that puberty has nothing to do with the obligation to keep Mitzvos starting at 13 years old. He says that the Halacha L'Moshe MiSinai is that a male becomes obligated to keep Mitzvos at 13 years old.
See here as well for some more sources.