Question: Why did G-d command the priests to eat the meat of sacrificial animals?

Answer: Thank you for your recent inquiry concerning Priests eating sacrifices.

In interpreting the sacrifices, I am following the symbolic methods introduced by Rabbi Samson Raphael Hirsch in his commentary on Leviticus. Should you wish to study it, this commentary is available in English. Rabbi Hirsch shows how the the sacrificial laws mirror psychological guidance.

Although the priests do not eat the “burnt” offering, they eat sin offerings and peace offerings. Rabbi Hirsch explains the symbolism: A sin offering is brought, say, if a person accidentally committed incest. This could plausibly happen in a poverty stricken society where a person comes home, the house is dark because of inadequate lighting, and the person has relations without waking his partner (even though the latter is frowned upon in Jewish tradition). If, as frequently happens in poverty areas, other family members are in the house – sisters, grown daughters – this person may inadvertently commit incest.

In such a case, the person brings a sin offering. He has committed a serious sin, but it was inadvertent. The priest eats this offering in a state of sanctity. This symbolizes that our physical needs should not just be done, but should rather be done in a priestly, regal manner. Indulgence should be done with a sacred approach.

In other words, the priest becomes a role model for indulgence to this person, and this helps prevent further sins.

The other sacrificial obligations of priestly eating can be interpreted similarly. For further details please read the source I alluded to above

Respectfully,
Russell Jay Hendel