Empathy is a big subject for Aspies, because we are told we lack it, and we really don't feel that way at all. Empathy is defined in two ways: first, its defined as understanding the emotional state of another human being; second, its defined as the ability to read the emotions of another human being without having to be told what they might be.
Aspies have problems understanding the underlying, subvocal communications of other humans. This has nothing to do with our ability to understand, and sympathize with, the emotions of others.
We have no problems reading the emotional states of animals. For instance, we often have extremely strong bonds with pets to whom we feel a closeness, just like everybody else.
This isn't something that we can't learn with help, and a lot of effort. Once trained, we can discern another person's emotional state quite well. Sometimes we get wrapped up in our own interests, and forget to check how other people are feeling.