Leviticus 12:7

and the priest shall offer it before the Lord and make atonement for her; then shall she be purified from either source of (her) blood. This is the law of the purification of her who hath borne a son or a daughter.

He will sacrifice both the lamb and the young pigeon or turtle-dove. Scripture employs brevity and thus does not mention the breaking of the neck of the sin-offering bird (for the kohen does not eat of the sin-offering when it is a bird).

he will make atonement for her as I have explained earlier

she will then have ritually purified her source of blood This teaches us that after childbirth she does not become ritually pure until the end of the enumerated days.

AND HE SHALL OFFER IT BEFORE THE ETERNAL, AND MAKE ATONEMENT FOR HER; AND SHE SHALL BE CLEANSED FROM THE FOUNTAIN OF HER BLOOD. Scripture is stating that she shall offer a ransom for her soul before the Eternal so that she shall be cleansed from the fountain of her blood, for a woman in childbirth has a troubled fountain and a tainted spring, and after she has completed the number of days of becoming clean [as explained above in Verse 4], or the days of the formation of the child, male or female,) — she shall then bring etc. she shall then bring a ransom for her soul so that her fountain should be stayed, and that she should become cleansed, for G-d, praised be He, “heals all flesh and does wondrously.”
Now our Rabbis have said that [the reason for these offerings is] that at the moment that she bends down to give birth she rashly swears [because of the pains of childbirth]: “I will no longer have relationships with my husband” [so as not to conceive again]. The main purport of this statement of the Rabbis is that since she only swears on account of her pain, and the oath is moreover not capable of fulfillment, because she is subject to her husband, therefore the Torah wished for her to atone for that which came into her mind [and therefore commanded her to bring these offerings]. G-d’s thoughts, blessed be He, are deep, and His mercies are bountiful, for it is His desire to justify His creatures.

וטהרה, she is ritually pure enough to be allowed to eat sacred foods. This is the interpretation of our sages in Yevamot 74. This is not so surprising, as a person undergoing the ritual rehabilitation process, if lacking only the official stamp of atonement, i.e. the sacrifice in question, may already eat sacred foods of the sanctity level of T’rumah, grain or its derivative given to the priest as a tithe. Sacred foods of a higher degree of sanctity may not be consumed by such a person until he has brought the requisite sacrifice.

וטהרה ממקור דמיה, “she shall be cleansed from the source of her ritual impurity.” What is the purpose of this verse? Even if the woman who had given birth had seen menstrual blood during all these days of ritual cleansing, she would be able to bring the requisite offerings at the conclusion of these 33 or 66 days after the initial period when she was out of bounds to her husband? Even if we were to assume that these words refer to the blood she saw while ritually unclean, there was no need to repeat the word דמיה, “her blood (her insides had not healed yet).” Her ritual purity does not depend on her bleeding or not bleeding, as she is not able to offer the sacrifices concluding her state of being considered a יולדת “birthing woman” even if she had not only stopped bleeding immediately after giving birth, but had experienced what halachah describes as a “dry” birth! The reason why the expression וטהרה, “she regained her ritual purity,” is in place is because until the expiry of these days and her having brought the requisite offerings, she cannot partake of food which is the residue of animals that have been slaughtered on the altar in the courtyard of the Temple, as well as agricultural products to be given to a priest, i.e. t’rumah.”