Most of the anti-realist positions focus on attacking that these sorts of moral statements, such as "x is wrong" or "y is the right thing to do" either intend to be statements of fact, are capable of being true or false, or both.
One problem that arises is the fact that morality doesn't seem to be the type of thing that can be established scientifically, that is, it seems to be in conflict with naturalism, the belief that facts can only be about the natural world, or that nothing real is supernatural and there are no supernatural facts. A particularly strong version of naturalism would physicalism, the belief that only the physical is real, and that everything is reducible/explainable by physical phenomena. Hence, you typically see hardline physicalists adopting moral anti-realist positions more often than people who hold other metaphysical beliefs. However, that's not always the case, as there are also many physicalists who are moral realists.
Why the discrepancy? There was a very inluential argument that seemed to commit moral realists to some very unintuitive results for many years called the open question argument, first formulated by G.E. Moore.
The argument attempts to establish the "open question" nature of moral questions, such as "is it good to do x?" by showing that no natural/physical/scientific terms are analytically equivalent (you cannot substitute them/definitionally interchangeable such as "bachelor" and "unmarried man") with moral ones, otherwise it would not make sense to ask "is x good?"
With minimalism about truth becoming much more popular, the open question argument has fallen much out of favor, and it was one of the strongest arguments putting a wedge in between moral realism and many forms of naturalism.
Essentially, if we accept that moral statements similar to the above can be true or false, then we have to admit that morals are real, since their existence is presupposed by the statement. Having either value requires them to be real, otherwise it could not be true or false.
"Colorless green ideas sleep furiously."
Is this statement true or false? It doesn't really make sense to say that this statement could be true. However, one might say, "well obviously it is false if it can't be true," and one might be forgiven for thinking that. However if this were the case, it would be a consequence that:
"Colorless green ideas are not sleeping furiously."
is true. That certainly doesn't sound right. Merely by admitting that there can be statements of fact with moral qualifiers, terms, or predicates which are not higher-order statements such as attribution, it commits one to some level of moral realism.
If you are interested in reading more:http://plato.stanford.edu/entries/moral-realism/https://en.wikipedia.org/wiki/Open-question_argumenthttp://plato.stanford.edu/entries/naturalism/