“feed on the sin of my people.” The word translated “sin,” chattah (#02403 חַטָּאָה) means both “sin” (Lev. 4:3) and “sin offering” (Lev. 4:8). Here in Hosea 4:8 it likely means both and thus is an amphibologia, a double entendre. The priests were living off of (feeding on) the sin of the people by literally eating the sin offerings of the people (“feed on” can also be translated “eat up” as in the KJV).
The priests were allowed to eat portions of many of the sacrifices. For example, the priest ate some of the grain offerings (Lev. 6:4-16, 18; 7:9-10), the sin offerings (Lev. 6:26, 29), the guilt offerings (Lev. 7:6), and the fellowship offerings (Lev. 7:28-34). The fact that the priests got to eat parts of some of the sacrifices is one reason why the altar of the Lord was also referred to as the “table” of the Lord (see commentary on Mal. 1:7).
We learn from Hosea 1:1 that Hosea wrote at the very end of the kingdom of Israel, after which it was conquered and deported by the King of Assyria (2 Kings 17:5-23). At the time of Hosea, the political and priestly system of the Kingdom of Israel was very corrupt; in fact, the whole priestly system in the Kingdom of Israel had started with priests who were not from Aaron and who led the worship of golden calves (1 Kings 12:28-32). The priests in the Kingdom of Judah were corrupt as well.
The second stanza of the verse, “and set their heart on their iniquity,” shows that the priests enjoyed eating the rich meat of the various offerings, and thus had no real incentive to work to help the people overcome their sin. The more the people sinned, the more food the priests got to eat. God originally gave the priests a share of some of the sacrifices as a reward for their labor, but now the priests had perverted the system and were happy when people sinned.
That the priest got to regularly eat meat already put them in a very privileged position in the biblical culture, because it was not common in those days for people to eat meat for a number of reasons: most people were poor, there was no way to reliably preserve the meat, and much of the meat that was eaten was of poor quality. Proverbs 7:14 gives us some insight into how attractive good meat could be in the biblical culture and how it could be used to help lure people into sin (see commentary on Prov. 7:14). The priests got the best meat in the land because the animals had been offered to God, but they were not thankful for that and instead looked forward to people sinning so they could have more.
One point that the Bible makes again and again is that if the heart of the person doing the sacrifice was not right with God, then the sacrifice was not acceptable to God (see commentaries on Amos 5:22 and Matt. 5:24). This should have been something that the priests were teaching the people and themselves as well.