If animal sacrifices did not atone for sins, why did God require them?
Why did sacrifices continue even after Yeshua's death on the cross?
Thanks for your questions about sacrifices.
It is hard to sum up the purpose of sacrifices in a few paragraphs, but I will try. For a more comprehensive answer, I highly recommend the Hebrews audio series by D. T. Lancaster.
Animal sacrifices do in fact atone for sins; the issue is the meaning and range of the term "atonement." Atonement does not directly equate with forgiveness.
Many people are under the impression that the Tabernacle existed for the sake of offering sacrifices. This is not the case, since it was entirely possible to offer sacrifices before there was a Tabernacle. Rather, the sacrifices were offered for the sake of having a Tabernacle. In other words, the sacrifices were a means to an end. The Tabernacle, as a place for God and man to dwell together in this world, was the end goal.
The Tabernacle and Temple provided a place for God's presence in this world. But God's holiness is so intense, it is dangerous for the non-holy to come in contact with him. The Bible speaks of God as a "consuming fire" (Deuteronomy 4:24, 9:3, Hebrews 12:29). Nadab and Abihu, when they brought "strange" (i.e., from a different, non-holy source) fire to the altar of incense, they were consumed by the fire of God. The details of the construction and operation of the Tabernacle must be followed to precise detail, so as to create an environment of holiness.
This presents a challenge, however, because of the inherent non-holiness of humans and our world. The whole point of the Tabernacle and Temple is the ability for God and man to be together. But when humans in a non-holy state enter the presence of God, they are consumed just as Nadab and Abihu were. Human non-holiness comes not only from sin, but also mortality itself. We remain in non-holiness as long as we are subject to sin and/or mortality.
The purpose of animal sacrifices as well as other Temple offerings and rituals is to provide atonement. The Hebrew word for atonement is kapparah, which literally means "covering." It is a buffer that provides a layer of protection between God's holiness and earthly non-holiness. It can be compared to a fire fighter's flame retardant suit that enables him to enter a burning building without being harmed. By achieving kapparah, the end goal can be accomplished: man abiding in God's presence without the risk of harm. But note that the atonement achieved here in the earthly Temple has no bearing on one's eternal judgment and destiny. It applies entirely to this life and the prospect of experiencing God's presence in the physical Temple.
The book of Hebrews explains that there are two Sanctuaries: the earthly Sanctuary and the heavenly Sanctuary. It draws a contrast between the two. Yeshua's sacrifice and priestly ministry operate exclusively in the realm of the heavenly Sanctuary. In the earthly Sanctuary he does not qualify as either a priest or sacrifice. The function of the heavenly Sanctuary is in regard to eternal life and the world to come. The earthly Sanctuary and the atonement secured there is limited in scope to this world and meeting with God in the physical Temple.
When Yeshua secured atonement for us in the world to come, he accomplished something amazing and of enormous significance. However, it applies only in the heavenly Tabernacle and that future world. Until that world is fully realized, the same rules continue to apply in a physical, earthly Temple. That is why the believers who met in the Temple in Jerusalem continued to participate in the sacrificial system.
I know this is a complicated issue, and I hope I communicated clearly.