Why did the priests eat the offerings?


You are to have the part of the most holy offerings that is kept from the fire. From all the gifts they bring me as most holy offerings, whether grain or sin or guilt offerings, that part belongs to you and your sons. Eat it as something most holy; every male shall eat it. You must regard it as holy. Numbers 18:9-10
            The consumption of the offering was twofold.  One, it fed the priests and their families who lived off the offerings.  The second reason was more spiritual, as the priest got rid of the sin the sacrifice represented and absorbed it into themselves.
            Jesus, as the high priest, absorbed our sins into himself and died because of them and for us.  The priest at the time of Moses symbolically represented the future Messiah who would do this once and for all.