I think a reasonable proxy for "care" is genuine curiosity about who a person is, how they came to be where they are, what makes them tick, coupled with a general desire for positive things to happen to them in their life.

If you believe that most people have things about them that are fundamentally interesting, you will put effort in to find those things, and you'll generally be successful in finding them. If you have a belief that most people are fundamentally uninteresting, you will not put in the work, and your beliefs will be validated because you will fail discover anything of interest in most people around you.

The act of earnestly wondering about a person is very flattering - this will help people open up to you and share their interestingness. If you can layer that with an additional thought of what useful thing you might be able to do for them, such as another person who might be able to help them, or a piece of advice you can give that might save them time, you will develop a reputation as a person one should talk to.

This then will increase your hit rate on being useful which is a nicely positively compounding cycle.