This ordination offering is very similar to the peace offering, which is also called the fellowship offering (Leviticus 3).
In this offering, the fat was burned as a gift to the Lord. The breast and the thigh were first lifted up before the Lord as a sign that they belonged to him (29:24–26). After that, they became the portion for the priests to eat (29:26–28).
In a regular peace offering, the person who brought the sacrifice also ate part of the meat. This meal was a fellowship meal shared in God’s presence (Leviticus 7:11–18). In this case, the priests ate the meal as part of their ordination (Exodus 29:31–34).