Nature is, in terms of most human codes of ethics, in a terrible state. Even the animals we most admire and feel kinship with, e.g., dolphins, dogs, elephants, kill without remorse, have little or no sense of any right to property, and often live in social structures where roles are defined, and enforced, by sheer power. Mating is often, by human terms, nothing short of rape. What it boils down to is that most of nature acts in an almost purely selfish manner, and it does so with no sense of judgment or regret.
Yet humans have not lived this way in a long time. Even when we do so, we generally have qualms about killing our own species. We have a sense that a person who has been robbed has been wronged in some way. It doesn't mean that we don't do these things, but we have a sense that, in doing so, we are engaging in lesser conduct. The question is why?
One school of thought, the general libertarian way of thinking, is that our superior intellect has taught us that reciprocal agreements to enforce certain rights vis-a-vis one another benefits all of us. This is the general idea of the social contract. Utilitarianism as well.
Another is that humans are different than the rest of nature by our very essence. That humans inherently are designed to balance the needs of the self with the needs of the other. While not necessarily dependent on the existence of a God, this belief is dependent on some metaphysical belief that transcends a mere quid pro quo arrangement and depends on a belief in objective ethical or moral norms.
The problem with the first school of thought it that it is based ultimately not on a sense of right and wrong, but on a sense of effective or ineffective. Killing is not probited becasue it's wrong, but because it is effective in keeping social order. Well, what if that analysis changes? What if one with power decides that killing those that oppose it is actaully more effective in keeping social order? While we might cringe at such a position, it is actually perfectly defensible from a utilitarian standpoint. More importantly, though, what if we lose consensus altogether as to what is effective and ineffective. With no objective criteria to rely upon, chaos is just around the corner.
I'm trying to say far too much in far too few words here, but the crux is that societies stand only when there is general agreement as to most fundamental principles. An appeal to objective factors, as in theism generally as well is a natural law type theory, makes it much easier to maintain that consensus. One can point to any number of sins committed by any organized religion, but the fact remains that theism -- in the West, Christianity -- has reinforced the basic messages of the dignity and worth of the individual, regardless of power or social status, the existence of objective ethical norms, and of the need to balance one's own needs with those of society. That underlying and unifying message has been an indispensible thread holding Western society together for centuries.
It is no coincidence that the abolitionist movement and the civil rights movement were driven in very large part by devout theists -- Christians as to both, and Christians, Muslims, and Jews as to the latter. Even the labor movement leaned heavily on the support of the American Catholic Church to get any traction. The belief in higher, objective norms that overrode the previaling notions of effectiveness of the time, made it easier for these transformations. That is not to say that religion, or theism, or even natural law is essential in reordering a society's principles, but it's hard not to see how the inability to point to higher truth than effiency can thwart such efforts.
We live just a few decades removed from a time when Christian values were virtually synonmous with societal values in the Western World, so, today's non-theists grew up immersed in a society largely influenced by Christian values and have retained most of those values. As that changes, however, there is no reason to believe that society will not change more radically toward a state closer to the "natural" state where power relationships govern not just in practice, but in principle as well. If we only act differently than the rest of nature because we "know better," what happens if we get the sense that nature was right, and us wrong, all along? The society that this would produce could be one unrecognizable to us today.
Again, I'm saying this all far too quickly, and I apologize for that, but I think HT's point about relying on intellect to elevate us from more base principles is a very important one.