Peter Hurford has some interesting questions for moral realists. So here are my answers; unfortunately I only have time to be quite brief at the moment.
(1) Why is there only one particular morality?
'Morality' is not the sort of thing that can be one or many; it's just an abstract common category. One might as well ask why there is only one particular humanity; that could mean any number of very different things. And there are several different things that might be meant if we take the question to be a metonymy. For instance, we might be asking why there are universal moral principles; to which we would have to reply that it's because some universal rational principles fall into the category labeled by 'morality' -- those governing whether it's rational to throw out all standards of rationality, for instance, or the principle of noncontradiction as applied to moral situations. On the other hand, the word 'particular' suggests that the question is asking why there is only one moral code, understood as including all moral principles? And the answer is that there isn't: application of universal principles to diverse particular situations will diversify particular moral principles. This divergence affects the answers to all the questions.
(2) Where does morality come from?
If we are talking about universal principles, at least some of them are simply intrinsically necessary, while others are necessary given certain common natural facts (e.g., that human beings need to eat in order to live, or that human beings seek to learn by both imitation and reasoning). If we are talking about particular principles, they are solutions to problems raised by particular kinds of circumstances, based on the general universal moral propositions and the circumstantial facts. Hurford also asks, "Are moral facts contingent; could morality have been different? Is it possible to make it different in the future?" And we can see that some aren't contingent and some are, and that some can be made different in the future. by changing circumstances, and others can't (and, it should be said, the two distinctions are different: a moral principle being contingent does not of itself imply that it could be changed in the future).
(3) Why should we care about (your) morality?
The answer is that it no more matters whether anyone cares about moral principles, assuming that's what's in view, than it matters whether people care about rational standards in reasoning, and for exactly the same reason. This ties in to the question of wordplay that Hurford raises. If someone were to change the meaning of 'rationality' so that it allowed 'accepting contradictions as true just because one feels like it', it wouldn't change any of the obvious problems with being 'rational' in this way, nor would it change anything fundamental, because the whole point of realism is that it's not a matter of how one defines words, but a matter of facts and necessities. Whether people care, of course, is just a matter of motivation; whether people care about abstract algebra is completely irrelevant to the question of what's true about abstract algebra, and equally irrelevant to the question of what a given person can and can't rationally believe about abstract algebra. Changing the definition of the words 'rationality' or 'morality' doesn't affect any of the actual facts, necessities, or possibilities.
Hurford recognizes the possibility of raising this point, but I suspect he is assuming the division between what is commonly called instrumental rationality and epistemic rationality, and taking this to be exhaustive. This, however, is an untenable dichotomy. If we have rationality concerning means, we can sum over all possible means to get rationality concerning ends: there will be ends that cannot be reached by any possible means. So there's a non-epistemic rationality that is not itself what people generally regard as instrumental rationality. We can do similar things starting from epistemic rationality. The distinction between the two is not exhaustive, and there has never been any good reason to think it is exhaustive; it's quite easy to start with with them and show that either alone or in combination imply forms of rationality that cannot be put into either category. And, of course, what I said about 'morality' applies to 'rationality'; these are abstract category-labels able to include multitudes of different things, not unitary things in their own right. Rationality is as extensive as reason itself; it cannot be arbitrarily chopped up into bits.
Someone who didn't desire "morality" at all would be quite literally insane: they wouldn't desire, directly as an end or indirectly as a means, to conform to rational norms, social standards, prudential reasoning about means, or their own aesthetic tastes, all of which at least partly fall under the label. Beyond that, we'd have to look at each sort of thing in its own right.