One crucial point of the argument of Richard Dawkins is that we do not need religion to explain the foundations of our morality. Scientific theories of evolution provide convincing and powerful accounts of how our ethical behaviors have evolved.
For example, origins of altruism can be explained by models of interaction between agents, where certain actions lead to the increase or decrease of utility for the parties concerned. Models suggest that "free riders" (those agents that take advantage of other agents' altruistic behavior) can certainly arise under certain conditions. However, they never become the majority. It is an obvious fact that a community consisting entirely of free riders is not sustainable. Under robust interaction conditions, altruistic behaviors naturally evolve, as they are certainly adaptive for the individual agents as well as for the community, increasing the overall utility.
Morality does not need to be explained by religious commandments, which the human beings are required to follow without questioning their origins. Even if we have a free hand in choosing what codes of actions we adopt, we can end up as reasonably altruistic, caring beings, with the bonus that we have the freedom to think through the foggy mists of unquestioned systems of beliefs.