When we hurt ourselves, why do we apply ice packs? Is it the same as taking tylenol for a fever? It seems like inflammation is useful when you have an actual injury and you wouldn't want to interfere with that process, no?
We sometimes want to stop excessive swelling. Although swelling is very natural and is telling you to stop moving around and rest, sometimes we need to move around.
Cold application (especially when repeated over time, or alternated with heat) increases blood movement/drainage in the area and can reduce the amount of inflammatory mediators that stick around. These mediators can stick around for longer than Mother Nature intended, and that's why cold application after the acute injury (into the chronic phase) can also be helpful.