It occurs to me that the real answer to this question can be found by analogy with natural selection. To see why this is, try turning the question around: Would it be possible for morality not to exist?
Try to imagine a society in which theft, rape, and murder were considered proper and acceptable behavior. Obviously such a society would not last very long. If you imagine a world in which some societies freely allowed such acts and some did not, then within a couple of centuries, only the societies of the second type would still exist. Notice that this is still true even if none of the societies in question have any religion. It's purely a matter of the inevitable consequences of behavior.
One could imagine a morality-free society reaching the logical conclusion that theft, rape, and murder should be discouraged purely because they are dangerous both to society and to the individuals within it, and decreeing punshments for that purpose; however, this alone would be of limited effectiveness. The police cannot be everywhere, and people of normal intelligence could figure out ways of indulging in forbidden behavior while reducing the risk of punishment to an acceptable level.
Such a society would be much less stable than one in which each individual was also discouraged from engaging in theft, rape, or murder by some form of internal inhibition, which is to say, a sense of morality. Notice that for the sake of this argument it doesn't matter whether that inhibition is a genetic trait or just a cultural feature that some societies happen to possess. The point is, societies which possessed this feature would be much more stable than societies in which those behaviors were accepted, or societies in which they were discouraged only by punishment; and so, over time, only societies of that type would survive, while societies of the other two types would disappear.
The actual roots of morality probably lie in the social organization of primates. Chimpanzees have various constraints on behavior within their social groups. For example, males fight each other for dominance, but restrain themselves so that these conflicts rarely lead to death or serious injury. This isn't due to a commandment from some chimpanzee Moses; it's because chimpanzee groups regularly fight each other, and any group in which the adult males mostly killed each other off would likely be wiped out by some neighboring group in which they did not do so; thus, only those groups which harbored the inhibition about internal fighting survived to transmit their genes (including the genes for the inhibition) to future generations. Most other chimpanzee behavioral inhibitions have obvious similar explanations.
The more elaborate morality of humans probably evolved from such primate inhibitions over time as our ancestors' intelligence slowly increased. As Charles Darwin said, "Any animal whatever, endowed with well-marked social instincts, the parental and filial affections being here included, would inevitably acquire a moral sense or conscience, as soon as its intellectual powers had become as well developed, or nearly as well developed, as in man."
But whatever the origin of morality, the important point is that human societies which did not have it would be at a disadvantage in competing for survival with societies which did, and so today we see only societies of the latter type.
Notice that this argument applies only to actual morality, and not to the hodgepodge of sexual taboos which, weirdly, are what religious people tend to mean when they use the word "morality". There are no human societies in which theft, rape, and murder are considered morally right, but attitudes about contraception, homosexuality, abortion, adultery, pre-marital sex, etc. vary wildly from one society to the next; any or all of those things may be considered crimes worthy of death in some cultures, utterly unobjectionable in others. This is because under most conditions, a society's attitudes about these things have only a marginal impact on its chances of survival, so the "natural selection" effect is not triggered. (Obviously a society in which all sexual activity was homosexual or all pregnancies were aborted would not last long, but societies in which homosexuality and abortion are accepted when individuals prefer them, can and do continue indefinitely.)
So the question of why we have morality is similar to the question of why we have sexual desire or fear of death; if our species didn't have such traits, we wouldn't even exist to be asking the question.