Halakhah for Leviticus 12:8

It is from the roots of the commandment [that it is] in order that she be aroused by action to give thanks to God, blessed be He, who saved her from the pangs of her birth, which is a miraculous thing. And also they, may their memory be blessed, said (Niddah 31b) that a woman rushes to swear at the time of her throes that she will no longer [have sexual relations with] a man. And therefore she requires atonement, and hence her sacrifice is called a sin-offering, and it is stated about her, "and the priest shall atone for her" (Leviticus 12:8).