Commentaire sur Le Lévitique 7:35

ואת משחת אהרן ומשחת בניו, the reward for being anointed; they receive the hide of the burnt offerings, the meat of the ordinary sin offerings and guilt offerings. They receive the breads of the תודה offerings as well as the thigh and chest of the שלמים offerings as well as the part of the minchah not destined for the altar.

זאת משחת אהרן, “this is the consecrated portion of Aaron, etc.” They have been consecrated as reward for performing these tasks. Another interpretation of the word: משחה: “to be anointed with that oil is a mark of distinction for them.” (compare Exodus 29,29 למשחה בהם)