Survival of the fittest is not the only basis for evolution - even Darwin was aware of that, but since then we have realised that there are many other mechanisms that are important. Morality is something that most of us (there are exceptions) have a similar idea of - this can be influenced by culture, but is inherent within us - its no coincidence that most cultures have similar sets of morals.
We evolved living in small groups. For such groups to survive, it was necessary to be socially coherent, trusting and protective even of old and weak who would still add numbers to the group. Groups full of selfish individuals would not survive. Groups who were compassionate and moral towards others in the group, but not to those outside the group would survive well (those who were completely altruistic would also not survive as other groups would take advantage of them). Hence we have the development of the typical human as being moral towards those they know and less so towards strangers. I think that describes most of us fairly well.
Of course in this current world of globalisation, all the world has become the equivalant of the small group that we evolved in. Because we are aware of the similarity between ourselves and people across the world, we have come to treat them like our family. Of course, not everyone is like this - there is still a lot of fighting from people who are still in small group mentality - extremists (religious or political) who want to protect their group at the expense of others. This is only natural.
And even within our social groups, there will always be anomolies - some true altruists may survive, promoted by our sense of group morality into being saints. There will also always be a group of people who do not possess morals - psychopaths (estimated to be 1% of the population apparently). These people can survive in small numbers - they can prosper knowing that generally people are moral and expect everyone else to be moral too, because its the majority thing. However, they can never dominate because their overall effect is too disastrous on society as a whole, even if personally they can be successful. Equally, they can never be eliminated because our societies have too many rich pickings for such a person.
This is the balance that has evolved. Computer simulations have shown that a balanced population will consist mainly of people who are neither altruists nor completely selfish, but reciprical altruists - we act altruistically in general, but if someone is selfish or unkind towards us (or at least we think they are) we stop being altruistic towards this person. True altruists are not survivors. Humanity will always be flawed in this way. Yes we can almost always trust in and reassure ourselves of the kindness of human nature, but this is not due to external moral values, God, or a humanist superiority - it occurs only because it was evolutionarily the most successful method of increasing the population.