One unspoken, selfish reason to care more about other people is that… you end up worrying less.
The thing is, most of what we worry about when we worry about ourselves is bullshit. We get anxious about speaking in public, even though no one in the audience really cares. We wonder what other people think about us (they’re mostly not thinking about us). We get upset for not winning an argument (as if someone was keeping score and our social status depended on it).
When you worry about a loved one (a friend, a family member, your partner) all of these bullshit worries fade away. It is as if we suddenly become more objective and selective in our concerns, and we focus on the important things: health, happiness, meaningful interactions. And you don’t have to make an effort to drop the secondary concerns, it’s just automatic. How often have you felt anxious about a friend’s work presentation? And it’s not because you don’t care about them – it’s because you know that the work presentation doesn’t matter. A simple fact that you then forget when you think about your upcoming presentation.
Obviously concerns about a loved one’s health and happiness can be daunting, especially when that person is unhappy or sick. But you have the exact same concerns when you worry about yourself, plus the bullshit worries.
Many people tend to believe that there is no rational reason to be altruistic and compassionate. They then deduce that we need religion and metaphysical beliefs to get people to be nice to each other. I think this idea is wrong for a number of important reasons, from game theory to evolutionary biology. I would just add a further reason here: caring more about others is worrying less, and hence rational.