How or why are we moved to feel compassion or empathy for another person?

These are cognitive, emotional and compassionate empathy. This page explains what is meant by each of these types if empathy. It also explains how and why it is possible to demonstrate one or more of the three types of empathy, yet still come across as uncaring.

Explanation:

It is important to note that feelings of distress associated with emotional empathy don't necessarily mirror the emotions of the other person. Hodges and Myers note that, while empathetic people feel distress when someone falls, they aren't in the same physical pain. This type of empathy is especially relevant when it comes to discussions of compassionate human behavior.