>the ability to share someone else's feelings or experiences by imagining what it would be like to be in that person's situation
It isn't a question of caring about people. It is a question of being able to put yourself in the shoes of a stranger with which you might not have anything in common. If you can do that, you will likely have general compassion for immigrants, the poor, the sick, minorities, LGBTQ+ folks, and really anyone who is being persecuted, oppressed, or unjustly burdened by something outside their control. That is fundamentally a more left leaning mindset.
If you need more direct experience (and that includes hearing a firsthand account from someone you are counseling) to engender that compassion, you are more likely to only extend this compassion to people who you share a lot with like your family, friends, and community (not just geographically), while people outside those groups wouldn't automatically be granted that compassion. This is fundamentally a more right leaning mindset.
The respective "radii of emapthy" are just different sizes.
[1] - https://dictionary.cambridge.org/dictionary/english/empathy
"In my experience, the right wing is always asking 'What about me?' whereas the left wing asks 'What about them?' And that, in a nutshell, is why I will always lean to the left."
-Source unknown
EDIT: alternately, you could argue that the left simply has a more expansive definition of "in-group" than the right does, with fewer litmus tests as to who is granted membership. i.e. "I don't care about their skin color / sexual orientation / gender identity / disability status, they're still human beings and therefore we're on the same team." But it might be a distinction without a difference.