The first question that arises in my modern mindset is why would a loving God direct them to kill animals as atonement for their sin or as a fellowship offering? Why not allow them to only have grain offerings or recite some kind of penance for their sin? That seems to resonate better with the modern notion of a God who is all love and no judgement.
But the Bible is clear from Genesis 3 that the price of sin is great. The first sacrifice was the one God made in order to clothe Adam and Eve after they had sinned. There was a price paid, in blood, for their sin. So the sacrifices made by the Israelites in the desert, like those made by Abraham, Isaac, and Jacob, were intended to bring to mind the high cost of sin.
After the fall, Adam and Eve's sons, Cain and Abel, also brought sacrifices. While some have suggested that Abel's sacrifice was accepted because it was a blood sacrifice, and Cain's was rejected because it was not, further study shows God's awareness of the heart attitude behind each offering. Both in Genesis and in Leviticus, as well as today, what we offer to the Lord is not a bribe intended to convince God to forgive us (and allow us to continue sinning without consequence as long as we provide regular offerings).
The sacrifice is intended to represent our repentance for sin, our recognition of sin's devastating impact on our relationship with God, and our sincere desire to put to death anything that comes between us and the Lord. Christ's sacrifice covers all of that so that now, instead of bringing a lamb, we look to the Lamb and consider the cost of our sin.