The Kohanim eat the Korban and those who brought the Korban get atonement.
Eating the Korbanos was an opportunity to make a Mitzvah a very part of our body. The Kohanim ate, filled their bodies with Mitzva, and the owners got atonement.
Nowadays, there is only one Mitzva that we eat, only one way to fill our bodies with Mitzvos. Matza on Pesach. Ahh Mmaatzaah!