On kin selection
I chose not to mention it as a possible explanation for our moral instincts, not because I don't think it's true or relevant, but because I don't think it's central. Specifically, it doesn't explain why we're so often nice to strangers.
Note, also, that using kin selection to explain morality or altruism implies that those behaviors are vestigial. See Dawkins's remarks below.
Here's Dawkins on the Waking Up podcast (episode 105, near the 33-minute mark), explaining altruism in a way that portrays it as vestigial (once but no longer adaptive):
I suppose a Darwinian view of altruism would go back to a time when we lived in small tribal groups, and there were two things about living in these small groups. One was that you were completely surrounded by kin: cousins, second cousins, siblings, nephews and nieces, and so on. So there would have been a Darwinian incentive to altruism toward anybody you meet, because anybody you meet is a member of your own village, your own tribe, your own clan. And the second thing, the other Darwinian engine of altruism, is reciprocation. And reciprocation largely depends on encountering the same individual again and again. And that again happens within the village, within the band, within the tribe. So there would have been a selection pressure toward within-group altruism.... And [now] we've moved out of our tribal past and moved into big cities, where we're no longer in tribal groups. But we still have the same rules of thumb, which work, and that is a good thing. We have a rule of thumb that just says, "In general, be nice to anybody, empathize with anybody." Because in the distant past, anybody would have been defined as your own tribe, your own clan, your own kin-group, your own reciprocation group.
[Transcript not perfectly verbatim; modified slightly for clarity.]
FWIW, I hope I'm not playing "gotcha" here. These were spontaneous remarks in a conversation that flitted across many topics, and I'm not sure whether they represent Dawkins's most carefully-considered views. In general, I have tremendous respect for Dawkins as an evolutionary thinker. I just wanted to show how these explanations (that imply morality is vestigial) are reasonably common.
In general, I'm not sure where to draw the line between (a) instincts that arose via individual selection and (b) behaviors that we learn from the environment via reinforcement learning.
Certainly we have some hardwired instincts, but we're also fairly general-purpose reinforcement learners. There's also a middle ground, which is to have prewired tendencies which are then accentuated or attenuated by what happens to us after birth. So we often talk about certain strategies being "embedded in our genes," but I think they're could just as easily be embedded in our environment (implicitly, in the form of a niche).
Further, I often wonder whether "individual selection" — or more broadly, "evolution" — is used as a crutch, enabling us to discuss self-interest at a safe remove. It gives us license to speak (and think) as scientists rather than as persons. It's just a lot easier to say, "We evolved to do [self-interested thing X]," rather than "We learn to do [X] because we're self-interested and [X] is good for us." A lot of great thinking has resulted from this mindset, but I worry that there's some sleight of mind going on here.
Selected quotes on group selection
(IMO all of these are wrong-headed.)
E. O. Wilson, The Social Conquest of Earth, p. 56:
An unavoidable and perpetual war exists between honor, virtue, and duty, the products of group selection, on one side, and selfishness, cowardice, and hypocrisy, the products of individual selection, on the other side.
Ibid., p. 241:
The dilemma of good and evil was created by multilevel selection, in which individual selection and group selection act together on the same individual but largely in opposition to each other. Individual selection is the result of competition for survival and reproduction among members of the same group. It shapes instincts in each member that are fundamentally selfish with reference to other members. In contrast, group selection consists of competition between societies, through both direct conflict and differential competence in exploiting the environment. Group selection shapes instincts that tend to make individuals altruistic toward one another (but not toward members of other groups). Individual selection is responsible for much of what we call sin, while group selection is responsible for the greater part of virtue. Together they have created the conflict between the poorer and the better angels of our nature.
E. O. Wilson, "Evolution and Our Inner Conflict":
Risking oversimplification, individual selection promoted sin, while group selection promoted virtue.
David Sloan Wilson and Elliott Sober, "Reintroducing group selection to the human behavioral sciences" (as quoted here):
An effective group-oriented society cannot be composed of individuals who are motivated solely by a calculus of self-interest. The external social conventions that make freeloading difficult are evidently necessary but not sufficient and must be supplemented by a psychological attitude of genuine concern for others; a direct calculus of group interest rather than self interest is essential.