The earliest source for treating the bar mitzvah derashah as transforming the occasion into a seudat mitzvah is Yam Shel Shlomo (Bava Kamma 7:37), which lists the day a boy becomes bar mitzvah and the feast held in his honor among the occasions counted as seudot mitzvah, alongside a brit milah, pidyon haben, siyum, and chanukat habayit.
Building on this, Mishnah Berurah 225:6 rules that there is a mitzvah to make a seudah on the day a son becomes bar mitzvah, comparing it to the day one enters the chuppah, and adds that if the boy delivers a derashah, the meal is a seudat mitzvah even if held on a different day than the bar mitzvah itself.
Arukh HaShulchan, Orach Chaim 225:4 and Kitzur Shulchan Arukh 61:8 both codify the same ruling in nearly identical language, explicitly citing the Yam Shel Shlomo as their source: the seudah held on the day the boy enters his fourteenth year is itself a mitzvah, and if the boy delivers a derashah, the meal retains the status of a seudat mitzvah even when detached from that day.
Responsa Benei Banim, Volume II 17:8-9 traces the Magen Avraham's citation of this din back to the Yam Shel Shlomo's own words, which ground the elevated status of the derashah-meal in a comparison to the seudat chanukat habayit: just as one who wishes to inaugurate his house with Torah and mitzvot rather than levity makes a seudah accompanied by a derashah fitting the occasion, so too the bar mitzvah meal becomes a seudat mitzvah through the boy's derashah.
Applying this halachah practically, Maamar Mordechai Hilchot Taaniyot U'Vein HaMetzarim 20:35 rules that a bar mitzvah seudah in which the boy delivers divrei Torah and which takes place on the actual day he turns thirteen is a seudat mitzvah permitting meat and wine, and he notes the Kaf HaChaim's citation of the Magen Avraham that even off the actual date, the boy's derashah confers that status.
Beyond the halachic framing of the meal, the custom of pairing the occasion with a derashah and communal recognition is reflected in Nitei Gavriel, Bar Mitzva, perek 16, which describes the bar mitzvah being marked on the following Shabbat with refreshments at the boy's home attended by the rabbi, relatives, and friends, at which the boy delivered a derashah, while Nitei Gavriel, Bar Mitzva, perek 4 describes a related custom of bringing the boy before an elder for a blessing that he merit success in Torah.
A different, non-halachic root for the practice of associating Torah words with a father's celebratory feast for a son appears in Shenei Luchot HaBerit, Shaar HaOtiyot, Kedushat HaAchilah, Emek Beracha 3:10, which reads Rabbi Akiva's toasts at the feast for his son in Shabbat 67b as aimed at stirring the sages and their students to speak words of Torah once the wine has loosened their hearts, rather than mere physical enjoyment.
