THE PAIN OF CHILDBIRTH
Upon examining the verse, “When a woman conceives and gives birth... she shall become impure... as during the days of her menstruant infirmity” — the section that discusses a woman who gives birth — we are reminded of the connection between a woman giving birth and pain (as punishment for a sin) recorded in parshas Bereishis: Hashem told Chavah (after she sinned by eating forbidden fruit from the Tree of Knowledge), “‘I will greatly increase your suffering and your pregnancy; in pain shall you bear children.” This establishes that every childbirth (which is the outcome of conception, “conceived and bore”) now happens in a manner of (your suffering and your pregnancy — the pain of pregnancy, and) “in pain shall you bear children,” “the pain of giving birth” and thereby contains a trace of the consequence for the sin of the Tree of Knowledge.
We can now understand why a woman who gives birth becomes impure (even if there is no sighting of blood) according to pshat: Just as the sin of the Tree of Knowledge causes pain to the birthing mother, it similarly imparts impurity.
This helps explain why a woman who gives birth must offer an atonement sacrifice: since every birth (of all women) contains a trace of the sin of the Tree of Knowledge, a new mother must bring a sacrifice of atonement.
[This is similar to Rashi’s explanation regarding the sin of the Golden Calf — “and on a day that I make an accounting, I shall bring to account — always, whenever I shall make an accounting of {the Jewish nation’s} sins against them, I will also bring to account against them a bit of this sin along with the other sins. No punishment comes upon Israel, which does not have some retribution for the sin of the golden calf.”]