In Leviticus 10:16-20, I understand that Moses was upset with Eleazar and Ithamar because they didn't eat the sin offering as they were instructed to. However, I don't see why Moses was satisfied when Aaron told him that they did burnt offering and sin offering on the same day and that's why they didn't eat the sin offering. What am I missing here?
To understand, keep in mind that Aaron lost two sons earlier this day because they violated the command of God (Leviticus 10:1-3). Thus, Moses was particularly jumpy about the sacrifices being done correctly because he did not want to lose any more family to God's wrath. "Then Moses made careful inquiry about the goat of the sin offering, and there it was-burned up. And he was angry with Eleazar and Ithamar, the sons of Aaron who were left, saying, 'Why have you not eaten the sin offering in a holy place, since it is most holy, and God has given it to you to bear the guilt of the congregation, to make atonement for them before the LORD?'" (Leviticus 6:17).
The law was, "The priest who offers it for sin shall eat it. In a holy place it shall be eaten, in the court of the tabernacle of meeting" (Leviticus 6:26). The problem is that eaten portion of the sacrifice was an act of joy and thanksgiving. "And there you shall eat before the LORD your God, and you shall rejoice in all to which you have put your hand, you and your households, in which the LORD your God has blessed you" (Deuteronomy 12:7). The worship offered was to be heartfelt. "I have not eaten any of it when in mourning, nor have I removed any of it for an unclean use, nor given any of it for the dead. I have obeyed the voice of the LORD my God, and have done according to all that You have commanded me" (Deuteronomy 26:14).
Thus, when Moses demanded to know why the sacrifice was not eaten, Aaron replied that he could not eat it with joy because of the recent deaths of his sons. Aaron thought it better to skip that portion of the worship than to offer the Lord less than his best or in a way that was against how God wanted it done. "And Aaron said to Moses, "Look, this day they have offered their sin offering and their burnt offering before the LORD, and such things have befallen me! If I had eaten the sin offering today, would it have been accepted in the sight of the LORD?"" (Leviticus 10:19). In other words, Aaron understood it was wrong not to eat the sacrifice, but he didn't want to commit a worse sin by offering a sacrifice to God that wasn't acceptable.
Seeing that God had not destroyed the violators as He had done earlier and seeing that Aaron's reason was sound, Moses left the issue alone.