You Don't Have to Show Empathy
Empathy means to feel into someone else's experience as if it were your own. It isn't a performance. You don't have to show it. You're wired for it already. The one thing it runs on is attention. Give another person your attention, their words, their face, the tone under the words, and something in you starts to feel along with them. That's not a technique. It's the equipment you came with.
So the work isn't learning to look like you care. It's being willing to put your own view down for a while and give their view your attention. Look at the eyes. That's where most of it shows. Give the person the quality of attention you'd want when you're the one trying to be understood.
A few plain things help. Say back what you heard, not your read on it, just the thing they said, so they know it landed. Then check: is that what you meant? Did I miss anything? You will miss things, especially in an argument or on a tender subject, and asking is the only way to find out whether they feel understood. Ask if there's more. That one is a gift. It gives them room to empty the whole storehouse.
When someone can feel that you're with them, through nothing more than the quality of your attention, they relax. They feel less alone in it.
None of it is a formula, even though I've just made it sound like one. Put your point of view down. Give them your attention. Most of the time, that's all it is.