Unlike some people, I'm not going to lie and claim that I have all the answers to moral questions. I can't even say that there's a easy, clear, and concise way to argue that there are moral truths and falsehoods. Sadly, the way that we teach kids the difference between facts and opinions tends to leave kids philosophically vacuous and saying that all moral claims are "just your opinion."
Still, I will start off by observing that the people who argue that there are no moral facts tend to figure that since there remains such stark disagreement about what is morally right and wrong that it must be entirely subjective. That falls apart the moment that you acknowledge that there's stark disagreement in everything, including the hard sciences. Just because we're not aware of a truth about the physical universe doesn't mean that the truth doesn't exist. The same goes for the metaphysical.
At some point, though, actions do speak louder than words. This is the part that we all have to realize.
A lot of people deny that there are moral facts but I've never met a person who behaves that way. In fact, some of the most vindictive, holier than thou, self-righteous people that I've encountered are the same people who will confidently say that there are no moral facts.
Okay, if you're absolutely certain that all moral values are subjective and equally valid, why did you spend the last two days debating moral values? Why do you vote? Why do you join PETA? Why are you so passionately opposed to the idea that there are objectively right and wrong answers to moral questions?
Really, I think it's just as self-defeating as the argument that an opinion can't be wrong. The moment that I say that I believe that opinions can be wrong, you have to argue that my opinion is wrong.
Finally though, I think that most of the people who argue that there's no moral truth are people who need it to be true because their own moral compasses are so badly broken. I don't think that it's necessarily so bad that a person who will say that you can never say that child rape is objectively wrong is saying that because he or she actually wants to do that. What I am saying is that people with an authoritarian mindset, people who believe in using violence against ideological dissents, people who believe that making their own lives better justifies making the lives of others worse have to come to that logical conclusion lest they must admit to being garbage human beings in there own minds.
Now, I do want to make it clear that there are ways that believing in moral facts can be just as dangerous as believing that morality just lives in each person's brain. Believing with absolute certainty that your gay son will go to Hell if he ever acts on his desires has lead to honor killings. Some of those beliefs have lead people to embrace tyranny and violence in the name of God. If you couple belief in moral facts with the belief that you have all the facts and you have everything right, that throws the system of inquiry when it comes to morality out just like the insistence that there are no moral facts.
I do believe that the healthiest view of morality is an understanding that there are statements about morality that are true and some that are false and that none of us has perfect knowledge. That way, we can at least hope to have meaningful discussions.