We cannot explain our moral experiences as purely subjective phenomena. Our moral experience includes the gut-level awareness that some things are really right and some are really wrong. Only a metaphysical commitment to belief in an amoral reality could quell that awareness.
If there are pure moral relativists who do not regard anything as really right or really wrong, it is because they have persuaded themselves that their metaphysics knows more than their moral sense does. Their metaphysical side has gagged their moral side, commanding it to “shut up, you stupid ninny, there is no such thing as right or wrong!”
But the moral side still wants to speak. If it hears of a gay man being murdered for being gay, it tears off the gag and screams, “that’s wrong!” The metaphysical side will be quick or slow to re-apply the gag, depending on how careful the person is to protect his or her metaphysics; for when the moral side screams, “that’s wrong!” that’s when the metaphysical beliefs are most in danger.