The source and reason behind age 13 being the age of Bar Mitzvah:

When a boy turns 13 years of age, he becomes obligated in Torah and Mitzvos. Until that age he is considered a Katan, and is Biblically exempt from keeping Mitzvos.

The sources in Poskim who mention age 13:

Admur Basra 4:2; 37:3; 39:1 [regarding Tefillin]; 55:6; 128:49; 199:9 [regarding Birchas Hamazon]; 271:7 [regarding Kiddush]; 616:8 [regarding Yom Kippur]; See Michaber O.C. 616:2; E.H. 155:12-15 [regarding Miun] and 169:10 [regarding Chalitza]; Rambam Shevisas Asur 2:11 regarding fasting on Yom Kippur; Ishus 2:10; Mishneh Avos 5:21; Mishneh Nida 5:6 p. 45b “At age 12 her vows become valid”; Shut Harosh 16:1; Rashi Avos 5:21; Orchos Chaim Tefila 73; Magen Avos of Tashbeitz Avos ibid in name of Midrash; Maharil 51; Birkeiy Yosef 55; Igros Kodesh 5:76; Likkutei Sichos 10:70; 15:291; Reshimos Choveres 19 and 21 and 59; Shaareiy Bar Mitzvah pp. 76-94; Encyclopedia Talmudit Vol 4 Erech Bar Mitzvah footnotes 1-4; Nitei Gavriel 1:1 footnote 1 in length; Shevach Yakar 38:1

The scriptural basis for age 13:

1) Some learn that the age of Bar Mitzvah is a tradition of Moshe from Sinai, and is not sourced in the written Torah. [Teshuvos Harosh Kelal 16; Maharil 51; See Likkutei Sichos 10:70; 15:291; Igros Kodesh 5:76]

2) Others learn that it is learned from the verse in Vayishlach [Bereishis 34:5] regarding Shimon and Levi who are called “Ish”, which shows that when one turns 13 years old he is called an Ish, and a woman, being that she has greater maturity, she is Bas Mitzvah a year earlier, at age 12. [Rashi Avos 5:21; Rashi Nazir 29b; Rav Ovadia Bartenura Avos ibid; Tosafos Yom Tov Avos ibid; Machzor Vitri Avos; ibid; See Likkutei Sichos 10:70; Igros Kodesh 5:76; Reshimos Choveres 21 in length]

The reason for age 13:

The reason that a boy becomes obligated in Mitzvos in age 13 is because at this age his G-dly soul completes its entry into the body. [Admur in Basra 4:2]

The reason a girl turns Bas Mitzvah a year before a boy: A girl becomes obligated in Mitzvos a year earlier than a boy is because Hashem granted extra understanding [Bina] to a woman. This means that her mental maturity is faster than a man, and she thus reaches the age of obligation, one year earlier. [Nidda 45b; Likkutei Sichos 11:331; Igros Kodesh 6:199; Toras Menachem 27^th Elul 5742 p. 2269; See Torah Temima on Bamidbar 30:4]