The ethics of care is a normative ethical theory - a theory about what makes actions right or wrong. While consequentialist and deontological ethical theories emphasize universal standards and impartiality, ethics of care emphasize the importance of relationships.
The justice view of morality focuses on doing the right thing even if it requires personal cost or sacrificing the interest of those to whom one is close. The care view would instead say that we can and should put the interests of those who are close to us above the interests of complete strangers, and that we should cultivate our natural capacity to care for others and ourselves.
You appear no to have reached the moral development stage of either. Your human development appears only to encompass the first stage:
1. Pre-conventional - Goal is individual survival
2. conventional Self sacrifice is goodness
3. Post-conventional Principle of nonviolence: do not hurt others or self
It seems you are behind the rest of the world in human development. We care because we want to and we can.