Fine, I'll make an attempt and I'll try and keep it brief, so then you can distort my point, claim I'm making an argument I'm not making and then shoot down that point. Enjoy!
Humans evolved to live in groups. I believe this is an uncontroversial point. As such we developed strategies for living in groups. Among those strategies was an ability to think about and understand morality. Tigers don't have a sense of right and wrong, that is fairly unique to us. (Though there are other animals who also display some semblance of that as well, but that's neither here nor there.) If we're going to live in large groups we need to understand how to behave around one another, however humans also developed the capability for abstract thought, which, coupled with this allows us to extrapolate morality beyond mere survival strategies for how to treat one another. Now the fact is, even if the ability to think about morals is innate, specifically the content of what those morals are can be highly variable. Throughout time human beings have developed many different codes of morality, which have had some similarities and some differences to one another, and which often reflect the structure and needs of the society in which they were developed. This would seem to indicate that rather than being innate, morals which appear in more than one different moral code worldwide are just the sort of morals which serve to preserve a society of nearly any form. Now this means there is also a lot of historical accident in the development of morality as well.
Now as for how this applies to the modern world, and a naturalist worldview. Well, for a naturalist there is no greater power to be served by worrying about morality, so there has to be something real, something I know exists. What about myself, I know I exist and I know what I want from morality. I want not to be harmed, I want to be allowed to pursue life as I see fit without imposition from other people (in so far as doing so does not cause harm to others). I live in a society that has a code of morality that governs how people behave. But as a member of that society I also have the power to influence what that code of morality is, particularly if I join a larger movement of people with the same goals (ie. the secular humanist movement, given that I'm also an atheist as a result of my naturalist worldview, it's a very good fit). So now we have a movement of people which uses the harm principle and the central tenet of "don't impose arbitrary morals on others" as the guiding principles of morality, and fight to bring overall social morality in line with that, because that's how I would like to be treated, and as a result how I would like to treat other people.
Does that make sense? This view differs from Dawkinsian reciprocal altruism in that I'm not arguing it's a biologically evolved response (or not entirely).
That being said, I do think that reciprocal altruism fails in some ways to explain ALL of morality, but it's also not as unscientific as you claim. It is absolutely backed up by some evidence and there are cases where this idea really does seem to be at play.