Ramban: It implies that the Mitzvah to sanctify the firstborn ¹ came into immediate effect, (and that they sanctified the firstborn in the desert - Bechoros 4b).

Seforno: It implies that every Bechor Adam needs to be redeemed - like other forms of Hekdesh. Otherwise, he is prohibited from performing any mundane actitivities. ²

Bechoros 19a: In Parshas Re'eh, we learn from the verse, "Kol ha'Bechor... ha'Zachar, Takdish" (Devarim 15:19), that a female is not considered a Bechor. ¹

Rashi and all the commentaries: It means 'that which opens the womb.'

Bechoros 3a: It implies that it is only a firstborn animal belonging entirely to a Yisrael that is subject to the Kedushah of a Bechor - to preclude one in which a Nochri owns a part. ¹

Bechoros 19a: Because we learn in Devarim 15:19 that the Din of Bechor is restricted to a male, our Pasuk adds "Peter Kol Rechem" to exclude the first male to be born after a firstborn female. ¹

Bechoros 47b #1: To exclude a Yotzei Dofen (a baby that is born by Caesarian section) from the Din of Bechor.

Bechoros 47a #2: To teach us that the firstborn son of a Bas Levi is not subject to the Din of Bechor, even if she is married to a Yisrael.

Bechoros 19a: To exclude the first baby that is born naturally after a Yotzei Dofen (a baby that is born by Caesarean section) - even though it is the first baby to be born from the womb. ¹

Bechoros 13a: To teach us that the firstborn fetus of a calf that one purchases from a Nochri, one belonging to a Yisrael that one sells (illegally ¹ ) to a Nochri, that which one shares with a Nochri, or that one receives from him or gives to him in a business deal, ² does not have the Din of a Bechor when it is born. ³

Mechilta #1: It teaches us that animals belonging to a Levi, like the Levi himself, are not subject to the Bechorah.

Mechilta #2: It teaches us that, just as one is obligated to look after one's firstborn baby for thirty days and then to give him to the Kohen in any location, ¹ so too, is the Din regarding one's firstborn animals. Additionally, a firstborn animal that is born after a Nefel, like a firstborn baby, is not considered a Bechor. ²

Rashi and Ramban #1 (above, in 12:30): Despite the inference, all the Egyptian firstborn died, whether of their father or of their mother. In the event that there was no living firstborn in the family, then even the oldest person in the house died. ¹

Ramban (ibid.) #2: It was indeed only the firstborn of the mother who died during Makas Bechoros, as the Pasuk indicates here.

Ramban to 12:30: A mother's firstborn will be verifiable, ¹ which is not the case regarding the firstborn of the father (especially regarding animals). ² Tehilim 78:51 supports this opinion.

Gur Aryeh (to 12:30) #1: At Makas Bechoros as well, the primary Makah stuck the maternal firstborn (as among such people, paternity cannot be clearly ascribed). Nevertheless, the Makah then extended to include anyone who could at all be referred to as a "Bechor." Refer to 12:30:2.2:1.

Gur Aryeh (to 12:30) #2: Conceptually, the sanctity of a Bechor stems from its being first to emerge into existence. ³ Paternity does not (immediately) bring something new into the world. Refer to 12:30:2.2:2 and its notes.

Mechilta: In spite of the Pasuk in Re'eh, "Takdish La'Hashem Elokecha" (Devarim 15:19), implying that one is obligated to declare one's firstborn animal a Bechor, it is Kadosh even if one does not do so. ¹

Oznayim la'Torah #1: The Bechoros belong to Hashem with regard to performing the Avodah in the Mishkan, and inasmuch as they either need to be redeemed (Bechor Adam) or brought on the Mizbe'ach (Bechor Behemah).

Oznayim la'Torah #2: It is to console Yisrael, not to worry about having to dedicate their firstborn to the service of Hashem, since they (indeed the whole world) belong to Hashem anyway.

Mechilta: Even though it is not crucial, it is a Mitzvah to declare one's firstborn animal Hekdesh and one will receive reward for doing so. ¹

Maharal (Gevuros Hashem Ch. 38, p. 143): Makas Bechoros will be mentioned explicitly only in the second Parshah (in 13:15). In general, in the first Parshah (Kadesh), the Mitzvah is to sanctify our firstborn - in order to remember to the salvation of the Jewish firstborn in Egypt. ¹ The next Parshah (v'Hayah) tells us to then redeem our firstborn (and to offer up our firstborn animals) - to remember to the smiting of the Egyptian firstborn.