The Torah strictly forbids the consumption of meat from animals that died of natural causes or were killed by predators. Since this rule applies to the entire nation, directing it specifically toward the priests raises an immediate question about their unique status.
The primary approach among commentators is that priests require a special, reinforced warning due to the nature of their Temple service. When bringing a bird as a sin offering, the priests do not perform a standard ritual slaughter. Instead, they kill the bird by pinching off its head. Under normal circumstances, this action renders the bird forbidden to eat. However, the priests are commanded to eat from this specific sacrifice. Because they are permitted to consume this offering, there is a risk they might mistakenly conclude that regular, non-sacred meat prepared in the same way is also permitted. Therefore, a clear boundary is drawn to emphasize that outside the bounds of the specific sacrifices, all improperly killed animals remain absolutely forbidden to them.
Another perspective links this specific warning directly to the elevated spiritual status of the priests. Because they constantly serve in the holy environment of the Temple, they must maintain a higher standard of caution and complete separation from impurity compared to the rest of the nation [רד״ק, צאינה וראינה, ביאור שטיינזלץ]. The focus is specifically on the act of eating rather than touching, because the carcasses of certain pure birds do not transfer impurity through physical contact, but only when their meat is actually swallowed [רד״ק]. Taking this strictness even further, some explain that the priests are warned against eating an animal that is merely close to death, as it shares the appearance of a forbidden carcass [מלבי״ם].
Beyond the daily laws of purity, a different approach applies this command to a distant, future reality involving the ultimate war of Gog and Magog. During times of war, soldiers are sometimes permitted to eat forbidden foods out of sheer necessity. According to this view, in the future conflict, the entire nation of Israel might be granted temporary permission to consume improperly killed animals to survive. The command is directed solely at the priests to establish that even during such an extreme national emergency, they will remain bound by their sacred restrictions and must preserve their holiness [חומת אנך, אהבת יהונתן].