Well, the trivializing gambit here would be to just say that "caring about another person" just means that your empathy circuitry causes you to feel pain when you observe someone in an unfortunate situation and so your desire to help is triggered ultimately by the desire to remove this source of distress.

I'm not sure how concern for anothers well being would actually be implemented in a system that only has a mechanism for caring solely about its own well-being (ie how the mechanism would evolve). The push for cooperation probably came about more because we developed the ability to model other the internal states of critters like ourselves so that we could be mount a better offense or defense. The simplest mechanism just being to use a facial expression or posture to cause us to feel a toned down version of what we would normally feel when we had the same expression or posture (you're looking for information not to literally feel the same thing at the same intensity - when the biggest member of your pack is aggressing at you you probably want the desire to run away or submit to override the empathetic aggression).

It's worth noting (for me) that this doesn't diminish the importance of empathy and it doesn't mean that I don't really care about others. I think that caring for others is ultimately rooted in self-centeredness but like depth perception is probably a pre-installed circuit in our brains (a type I system) that we can't really remove totally without radically modifying the hardware. Caring about another person is as much a part of me as being able to recognize their face. The specific mechanism is only important when you're trying to do something specific with your caring circuits (or trying to figure out how to emulate them).