In the first place evolution doesn't aim towards anything so there is no expected outcome in evolution. 'Fairness' is obviously a trait that selects for survival in a social animal. The complexity of human behaviour is that it is rarely just about nature or nurture but is usually a blend of the two.
A good example is walking and running, which is an innate skill. We don't learn to walk. We are just born prematurely because of our huge brains so we have to be looked after by parents and family until the muscles are strong enough to stand upright and walk. But that basic ability to balance and walk upright can be developed through training to make us run faster or even walk on our hands, walk tightropes etc.
'Fairness' is a basic concept that is widely accepted and understood in all human societies as far as I'm aware. It is then overlaid by society and in groups beyond a simple tribe or family codified as moral or religious laws. That's why 'fairness' in some societies includes slavery and the oppression of women, LGBT people and people of other races. Religious 'morality' can include stoning, slavery, murder, rape, genocide and all the other ridiculous nonsense in Leviticus that are not acceptable (or sane) in a secular society. It's arguable that in the UK and other modern westernised countries, religious morality has been more heavily influenced by secular thinking than the other way round.
Walking is a good example. It just so happened that at some point in our evolutionary past our ancestors stopped running on four feet and started using two feet instead. Just like, at some point in the past, a different mammal developed wings, and some other mammal developed fins. All of these adaptions were better suited to their immediate ecological context than other adaptions and therefore were fitness-enhancing. Thus, relative to their context there was a fitness advantage for ancient apes to start walking.
But, having a relative advantage for adopting some behavior, does not make that behavior the objectively right thing to do. It would be absurd to say that it is objectively better to walk than to fly or to swim. The behavior is only advantageous given the relative context of the organism. This applies to every behavior that is developed instinctively (i.e. not learned). So if we have instincts for compassion, altruism, empathy etc etc we developed them only because our ancestors that developed this trait had an advantage over those ancestors that did not develop this trait. (As Corkboy pointed out above, developing compassion, altruism etc were probably evolutionary stable strategies - though, interestingly, our instinctive traits of compassion seem to only include members of our in-group rather than all moral actors, human or otherwise).
The problem however is that having an instinct for compassion, just because it was a relatively successful strategy in our past, is not what most of us mean by morality. Suppose you hear about a murder. You may have an instinctive gut reaction against the act. However, to really call ourselves moral, the gut reaction can't be enough. It cannot be that we oppose murder just because it was advantageous for our ancestors to do so. After all, it was advantageous for our ancestors to walk and not to fly. That doesn't mean it's wrong to fly even if it is not a behavior we adopted. To say that it is wrong to murder we need more than just our historical aversion to it. Instead of thinking to ourselves "murder is wrong because we just didn't evolve to murder", we think "murder is wrong because it is immoral". In other words, our aversion to murder comes from a rule we think applies to ourselves. We don't murder just because it is the wrong thing to do, independently of any instincts we may or may not have with regards to murder.
So the question is, where did these rules come from? And, to go back to the discussion above, do animals think in terms of rules? On the latter question I think it is debatable. Does a chimp avoid murder just because they just happened to evolve an aversion to it or do they avoid murder because they recognise a moral truth? I don't think the latter is all that plausible but may be true. It definitely isn't true once we get further down the evolutionary tree. I would need an incredible amount of evidence to prove that an ant grasps a moral truth when it behaves altruistically - rather than just unconsciously behaving as it has evolved to do, no different to it walking or building a hive, behaviors that carry no moral weight whatsoever. If I want to be a moral actor I have to behave in a way that I think follows whatever moral rules I give myself, not behave in a way that I've evolved to behave without considering the moral weight of my behavior. To do what we have just the instincts to do is not to make a real choice to be moral at all, it is just to behave without consideration. After all, I instinctively am attracted to sugary foods and instinctively laugh when I find something funny and instinctively avoid walking too close to the edge of a cliff. I don't attach any moral weight to these behaviors. However, when I instinctively want to tell the truth, I attach moral weight to this behavior because I recognise that this conforms to a moral rule I want to follow. That is why the instincts we have gained from evolution are not enough to make us moral creatures -what makes us moral is our self-recognition as moral, not the instincts we just happen to have (which are no different to walking, laughing, having a propensity to learn languages etc, behaviors that have no moral weight).
On the former question from above, (where did these rules come from?) there are a couple of options. Either there are real moral facts in the world, i.e. it is an objective fact about our reality that moral truths exist (somehow) and that these moral truths include things like "never torture a child" and somehow our minds can grasp these rules (I take it that Bob Sacamano holds this view). Or else, these rules are mere conventions/ fictions that we have developed because we realise that it is better to behave as if we are moral because it conforms with our instincts and gives us an easy time forming societies. I think Daniel Dennett holds this latter view. Basically, it is an error theory view about morality such that there are no real moral truths out in the world at all, genocide and torture aren't really wrong in the universe outside of human concerns, the universe has no moral nature a all and is indifferent. Some people believe this, other people find it repugnant and want to be moral realists. These people may think there really is something at stake when we talk about what the moral facts in reality are - it is always everywhere wrong to torture children and this fact is part of the fabric of reality and not just a fiction. Giving up moral realism is hard because then we have problems like explaining how or why moral progress happens (such as overcoming slavery). If there are no real moral facts out there in the world, and the universe is indifferent, then slavery isn't really right or wrong, it is just wrong according to the self-imposed rules of the people that oppose it or perfectly acceptable according to the self-imposed rules of the people that support it. In other words, without moral realism we have the issue of how to decide which rules are better.
But, in amongst all of the above, we should separate out the two issues I'm discussing. Whatever you think about the ontological status of moral rules, it is a separate issue from whether evolution can give us those rules. I don't think it can. All evolution can do is instill behaviors in us that are reinforced over generations. To be moral however is not just to behave some way, but to attach moral weight to that behavior. Just doing something out of instinct cannot have moral weight because it is not done for the right reasons. It's not enough to behave in a moral way because we can program our robots to behave in a moral way (see for example self driving cars that are programmed to uphold passenger safety before everything else). But what it takes to be moral is not just the behavior but the self-conscious acceptance of a moral rule (something that a robot cannot do and perhaps only the smartest animals can). Evolution gives us behavior, it doesn't give us a moral compass.
