This nature is best explained in the New Testament books that deal with the Old Testament sacrifices. Let me give you some verses.
Hebrews 10:4 states, "For it is not possible that the blood of bulls and of goats should take away sins." This shows that the animal sacrifices could not completely remove their sins.
Hebrews 10:11-12 states, "And every priest standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins: But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God." This passage contrasts the temporary effect of the Old Testament sacrifices with the absolute sacrifice of Jesus Christ.
Hebrews 7:19 reads, "For the law made nothing perfect, but the bringing in of a better hope did; by the which we draw nigh unto God." The sacrifices were a part of the law and this verse says that they made nothing perfect. But the better hope of Jesus Christ perfected our salvation.
Hebrews 9:13 explains the purpose of the sacrifices: "For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh." They were given for the sanctifying of the flesh, not the absolute salvation of the soul.
Now, why did their flesh need to be sanctified? This was necessary because God came to make His dwelling with the Israelites. Leviticus 16:16 states, "And he shall make an atonement for the holy place, because of the uncleanness of the children of Israel, and because of their transgressions in all their sins: and so shall he do for the tabernacle of the congregation, that remaineth among them in the midst of their uncleanness." This and other verses emphasis the importance of being holy because of the presence of God among them.
The sacrifices covered their sins but they did not completely do away with their sins. Consider this entire passage:
Hebrews 10:1 For the law having a shadow of good things to come, and not the very image of the things, can never with those sacrifices which they offered year by year continually make the comers thereunto perfect.
2 For then would they not have ceased to be offered? because that the worshippers once purged should have had no more conscience of sins.
3 But in those sacrifices there is a remembrance again made of sins every year.
4 For it is not possible that the blood of bulls and of goats should take away sins.
The reason they had to keep coming back is because the sacrifices did not entirely do away with their sins.