“a sin offering.” The reason for the sin offering is not specifically stated in the text. It seems clear that the act of sexual intercourse was not a sin (cp. Lev. 15:16), nor was the act of conception or birth. Also, it is worth noting that the animal required to expiate the sin was the least expensive of the acceptable blood sacrifices: a pigeon or dove. It has been suggested that the sin was what prevented the mother from entering the courts of the Temple during her uncleanness, but that explanation does not seem to fit the facts, because the act of sexual intercourse also prevented both the man and woman from entering the Temple but was cleansed only by bathing in water, not by a sin offering (Lev. 15:16-18).
It is apparent from the scope of Scripture that mankind has an inherent sinfulness, indeed, a sin nature that has been passed down from Adam, that must be atoned for. Thus, we see times in the Law when there was not a specific sin, but rather some sort of outward manifestation that in some way showed or recalled the innate sin of man, that called for a sin offering. For example, a sin offering had to be made for the priests before they were anointed to serve as priests, not because of any specific sin, but simply to be clean in the eyes of God. Here in Leviticus 12:6, the act of childbirth recalls the sin and curse of Genesis 3:16, that the woman would have travail in childbirth, and thus manifests the human sin nature, so God commanded that a sin offering was appropriate.