It illustrates how the death of Jesus Christ dealt with the root of sin. When God, who is holy, looks at us to judge us, His first reaction would be to immediately destroy us, since our sinful nature is repulsive to Him. The sin offering element in Jesus' death atones for our sinful nature. This interpretation is supported by the following:-
The sin offering is in view in the testimony that Jesus got from his cousin, John the Immerser: "The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin [singular, not plural] of the world." - John 1:29
Similarly, Isaiah wrote: "Jehovah has laid upon him the iniquity [singular] of us all" (Isaiah 53:6)
The Bible says that the crucifixion of Jesus Christ was a momentous event in history, because when he suffered and died, he paid the price for the sins of the whole world. During the three hours of darkness, God heaped His wrath on Jesus. God treated Jesus like we sinners deserve to be treated so that He can treat us how Jesus, His spotless, beloved Son, deserves to be treated.
If this is so, it is only fitting that God would institute animal sacrifices that would illustrate different facets of Jesus' death. Testimonies to how Jesus would fulfill or fulfilled these animal sacrifices are scattered in the Bible, written by various authors separated from one another in time by centuries.
Historically, animal sacrifice been an integral part of all ancient religions. This is consistent with the Bible's claim that God instituted animal sacrifices at the start of history (but inconsistent with evolutionary theory - if cultures developed separately all over the world shortly after homo sapiens spread out from Africa, there is no reason why all of them should have become religious, much less why all of them should think that killing animals serves a spiritual purpose). However, the Bible is the only religious book which records the institution of various types of sacrifices in painful detail and also how these sacrifices were fulfilled in One Grand Sacrifice that is the basis for sinful people to have access to a sin-hating, infinitely holy God. This is evidence that the Bible is indeed God-given (as it claims to be), and that turning to Jesus Christ is your only hope to avoid God's wrath.