Like many things empathy is a skill, and requires practice. But what and how to practice are tricky questions that require an understanding of how empathy works.
People commonly confuse empathy with sympathy but they are different.
Empathy is different to sympathy in that when we sympathise we feel warmth, sadness or other reactions to the person's situation or emotional states. Empathy is the experience of actually feeling what the other person is feeling, and communicating that to them through words or deeds.
Interestingly, it seems there is even a brain cell or "neuron" dedicated to helping us feel what others feel. Human beings (and some higher apes) have the ability to learn from observing others. When we observe others our brain responds like we are actually doing the activity ourselves. Parts of the brain that are active when we're observing and rehearsing actions in our mind are called "mirror neurons" and they're an important part of how we empathise as well.
Which leads us to the first, and most important tip when it comes to being more emapathic: We have to increase our own level of comfort with emotions in general. Because empathising involves tuning into the emotions of others, we have to be open to actually feeling it, and individually we all have varying levels of comfort with emotions.
There are also some very practical things you can do, and most of them involve paying attention in a very deliberate way. Mirror neurons seem to require both proximity and visual cues to actually work. So make sure you're not distracted (put down that phone!).
• To be really empathic we have to feel it. If you find yourself switching off, ask yourself why might this emotion be hard for me to feel?
• Sit facing the person. Make eye contact and try to really concentrate on looking at them, observing their words, posture and voice tone.
• Avoid distractions, don't try and watch TV, be on your phone or listen to the radio at the same time. If you're listening to someone, just listen.
• As discussed in a previous column, validate. Validating what we are picking up via empathy is a great way to let person know you're (at least trying) to empathise. You don't have to get it "right". Often just a "close enough" attempt is enough: "I can understand you feel angry, I would too in that situation..."
• Avoid offering solutions. Not only is this generally (accidentally) invalidating, really listening and emapthising is "doing" something, and a lot of the time actually "fixes it". At the very least it will improve your relationship, and create more connection and closeness.