Because he "feels anger," rather than "feeling angry." It is not reflexive to feel anger because he is actively feeling something (anger); it's a direct object of the verb. When he is feeling angry, it is reflexive upon himself. He is "feeling anger on himself," causing him to be in the state of angry.
On czuje złość, ale on czuje się zły, even though the latter means "he's feeling bad."
This is hard to explain. I hope it makes some kind of sense, since English doesn't use this type of construction.