Why does the Torah break off the narrative here, to discuss the lineage of Reuven, Shimon and Levi?
Rashi #1, Ramban, R. Bachye #1 and Rashbam #1: In order to present the Yichus of Levi, on account of Moshe and Aharon (the two redeemers ? R. Bachye). 1 And it would not have been befitting to begin with Levi, 2 so it begins with Reuven.
Rashi #2 (citing the Pesikta Gedolah) and Rashbam #2 (citing the Mechilta): The Torah makes a point of presenting the lineage of Reuven, Shimon and Levi, to demonstrate their importance, since Ya'akov berated them before his death. 3
R. Bachye: Since the end of the Galus was imminent, and Hashem was about to redeem Yisrael, the Torah lists their Yichus, to distinguish them from the (Tum'ah of ? P'sikta Zutr'sa) the Egyptians. 4
Seforno: Because all those listed, were appointed leaders of the generation of those who came down to Egypt. 5
Oznayim la'Torah: In case one asks as to why Malchus and Kehunah were given to Moshe and Aharon respectively, and not to someone from the tribe of Reuven, who was the firstborn, or from Shimon, who was born second, the Torah presents the names of the members of all three tribes to demonstrate that there was nobody in Shimon and Levi who could compare to Moshe and Aharon - which explains why Hashem chose the tribe of Levi. 6
Ramban and R. Bachye: ... thereby conveying the impression that Levi had now become the firstborn, in honor of Moshe.
Rashbam: And it makes a point of mentioning Korach, the sons of Uziel and Pinchas, so that, when the Torah discusses them later we should know who they are.
It is not clear however, why the Torah stops at Yehudah. The continuation of R. Bachye's explanation suggests that his main answer is the one quoted in answer #1.
See Seforno, who explains why the Torah lists only the sons of Reuven (the B'chor) and Shimon; but even the grandsons of Levi - who was the last of Ya'akov's children to die, and whom Levi, Kehas and Amram also educated to learn and to teach - Refer also to 6:16:2:2.
Oznayim la'Torah: Moreover, the list of B'nei Levi includes a number of people whom the Torah will record later that they did not die - such as the sons of Korach, the sons of Aharon and Pinchas.
"Eileh Rashei Beis Avosam". What are the implications of the word "Eileh"?
R. Bachye: With reference to 6:14:1:3 ? "Eileh" disqualifies the Egyptians (who remained with their Tum'ah). 1
See R' Chavel's footnotes.
QUESTIONS ON RASHI
Rashi writes: "Since [the verse] has to tell us the Tribe of Levi's pedigree until [reaching] Moshe and Aharon... it begins [writing] the lineage in the order of the [tribes'] birth, starting with Reuven." But why not start with Moshe and Aharon themselves - e.g., 'Moshe and Aharon were the sons of Amram, son of Kehas, son of Levi'?
Gur Aryeh: The Pasuk emphasizes that when Hashem selected a redeemer, he picked no one from the tribes of Reuven or of Shimon. Levi was third-born, and therefore predestined to father the redeemer. 1 Nor did any other member of the tribe of Levi qualify - only Moshe and Aharon.
Maharal (Gevuros Hashem, beg. Ch. 31, p. 119): Conceptually, a third entity is most apt to achieve completion. 2 The number 1 is not yet a count; and 2 is an even number which must share its status with its pair. 3
For more about the unique status of Shevet Levi according to Maharal, refer to 5:4:2.1:1, 5:4:3:6, as well as 2:1:2.1:1*.
See Maharal (Gevuros Hashem, beg. Ch. 54, p. 235) regarding Yaakov's status as the third chosen member of the Avos. Also see Maharal (Derush Al ha'Torah, printed at the end of Be'er ha'Golah, p. 24) - A third entity can stand apart from (mere) physicality. Portrayed on a graph, a third point is a median, in contrast to the prior two points, which are extremes (also refer to Bereishis 49:33:4.1:2 and its note). The Midah that this illustrates is 'Yosher' - honesty and justice. Maharal (Tif'eres Yisrael Ch. 11, p. 37) - Moshe - who was a third-born as well - exhibited this trait of Yosher in his formative years (see Shemos 2); when he felt his brethren's suffering, killed the Mitzri, confronted a Rasha, and rescued the daughters of Yisro. Refer to 2:11:2.03:1.
Maharal (ibid.): The term "choosing" applies to selecting one out of no less than three (because one out of two is half, which is almost a majority).