So what is the argument? The problem is not that atheists don't have morals but that they don't have a source for those morals that is greater than themselves. They can base morals on their feelings or on their reason and end up with a moral code that is pretty good. The problem is it is something they have created and therefore they can change it at any time. If they convince themselves that something is OK then there is no standard that someone can point to and show them they are wrong. The classic example is Hitler and genocide. This is just because it is something that everyone agrees is immoral. What do you say to a person who has honestly comes to the conclusion that genocide is a good idea? Is their opinion of what is moral just as valid as yours? You can question their motives. You can question their logic. They can question your motives and logic. But at the end of the day they can just dismiss your argument and simply say they disagree.
So why does this matter? Because humans are capable of skewing their reasoning. What is known as rationalization. Here is Thomas Merton:
I think that if there is one truth that people need to learn, in the world, especially today, it is this: that the intellect is only theoretically independent of desire and appetite in ordinary, actual practice. It is constantly being blinded and perverted by the ends and aims of passion, and the evidence it presents to us with such a show of impartiality and objectivity is fraught with interest and propaganda. We have become marvelous at self-delusion; all the more so, because we have gone to such trouble to convince ourselves of our own absolute infallibility. The desires of the flesh–and by that I mean not only sinful desires, but even the ordinary, normal appetites for comfort and ease and human respect, are fruitful sources of every kind of error and misjudgement, and because we have these yearnings in us, our intellects (which, if they operated all alone in a vacuum, would indeed, register with pure impartiality what they saw) present to us everything distorted and accommodated to the norms of our desire.Moral reasoning is one place where our intellect bumps right against appetites and desires. Atheist morality makes the assumption that reason will win that encounter. At least if we can put aside religion then human reason will be able to follow what logic tells him is right and not be hopelessly skewed by our passions. It isn't true. The most intelligent alcoholic will fall for the silliest line of reasoning if it justifies his drinking.
So what is the solution? People need some reason to overrule their own opinion. Sometimes the state law serves this purpose. But the laws can be immoral. Then there are cases like the Hitler example where the person has the power to rewrite the law. So we need some higher law. Like a law that says it is always wrong to break your marriage vows. It might come from religion or maybe just your sense of some moral absolutes. Then no matter how many interesting arguments for adultery cross your mind you dismiss them all because this law trumps your reason.This is good because those arguments are going to be skewed by your passions. But if you don't believe in moral absolutes then your chance of succumbing to bad reasoning is a lot higher.
The truth is that most atheists are moral because they have some unquestioned moral absolutes. This is a good thing. It is not a rational thing because they don't have a logical reason for believing these principles trump all logic. Yet they are there and they are typically true. Principles like honesty, hard work, the pursuit of truth no matter the cost, respect for other people, etc. The reality is most atheists were raised in a society formed by Christian morals. So they have picked up these ideas and have them so deeply ingrained they don't even know they are there. But how long will that be true? As the Christian world and life view becomes less and less dominant in the west how long will we see atheists displaying these values?