Many people appear to believe that ethical truths are culturally contingent in a way that scientific truths are not.
Harris surely means many nonreligious people, not many people. Religious people believe in absolute ethical truths handed down by God. Regardless, to use the phrase "ethical truths" is to assume that such a thing exists. But if ethical judgments reduce to emotional reactions, then the attribute of truth/falsity does not apply to moral judgments, anymore than it applies to any feeling. We can sensibly speak of a perception or belief as a true or false representation of the world. But one cannot sensibly say that a state of happiness in true or false.
Indeed, this loss of purchase upon ethical truth seems to be one of the principal shortcomings of secularism. The problem is that once we abandon our belief in a rule-making God, the question of why a given action is good or bad becomes a matter of debate. And a statement like "Murder is wrong," while being uncontroversial in mot circles, has never seemed anchored to the facts of this world in the way that statements about planets or molecules appear to be.
And this is because "Murder is wrong" is ultimately grounded on an emotion, a feeling of revulsion and loathing, not upon a factual description. People mistakenly sense that ethical pronouncements such as "Murder is wrong" have the same status as facts such as "Water boils at 100 degrees C" for three reasons. The immediate reason for equating moral statements with descriptive truths is the very strong feelings people have about behaviors such as murder. If you feel very strongly that something is a truth, then you are likely to believe it is a truth. Second, one's sense that a moral judgment represents a truth is reinforced when one's community shares the same strong moral intuitions. Third (and this is a more indirect and less conscious reason), presenting a moral claim as a truth rather than a feeling gives people more leverage when they are attempting to persuade others to comply with their feelings of morally desirable behavior. And that is the point of making public moral claims. If I tell someone, "You must fall at an accelerating rate of 9.8 meters per second per second," they cannot argue with me if they know their physics. It is the truth. If I tell someone, "You must not murder," the injunction may appear to be more objective and have more force if presented as a truth, a natural law, than if I say "You shouldn't murder because it makes me feel icky."
The confusion between moral injunctions and natural laws is reinforced in our language by double meanings of "right" and "wrong." On the one hand, "right" can be used to describe a truth, e.g., "4 is the right answer to 2 + 2." (And "5" would be one of an infinite number of wrong answers.) On the other hand, "right" refers to socially desirable or proper behavior, as in "Sharing with the less fortunate is the right thing to do." This linguistic ambiguity makes moral injunctions ("Do the right thing!") appear to be referring to necessary truths rather than mere, conventionally desirable behavior.
According to the Swiss psychology Jean Piaget, confusing moral rules and natural laws is a normal stage of cognitive development in children, a stage he called moral realism. Although Piaget claimed that adults have moved beyond moral realism to a stage of moral subjectivism (where rules are seen as a matter of working consensus), the widespread use of phrases such as "ethical truths" indicates that many adults are still exhibiting moral realism. The pervasiveness of confusing moral and natural laws may indicate some adaptive function for moral realism.
The problem, in philosophical terms, has been one of characterizing just what sort of "facts" our moral intuitions can be said to track—if indeed, they track anything of the kind.
Moral intuitions do not reflect facts about the world. Rather, they stem from emotional dispositions toward behaviors that helped perpetuate genes in the past.
I have many more comments on Harris's chapter, A Science of Good and Evil. This was just a commentary on the second paragraph of the chapter.