The Lord said unto Moses that the woman upon giving birth was unclean for a period of 40 days for giving birth to a man and 89 days with a maid. This is a reminder of the original sins of Adam and Eve. In the name of this sin, the woman is to delay her purification. The purpose here is to reflect on the fallen nature of man and how our salvation comes from God and Him alone.
With a sacrifice of a lamb and a pigeon, she could restore herself in His name. The lamb was the burnt offering and a pigeon the Sin Offering. Recognizing the sin of man and confessing them to God is the purpose of these sacrifices. Did you notice how this whole chapter is about recognizing God and His majesty as well as our shortcomings as man and woman? Give God the Glory and proclaim Him over all. Receive Him into your heart through the Holy Spirit.