I'm not a naturally empathetic person, but people often rate me highly these days. All of my tricks come from Carl Rogers' book "Active Listening", where he teaches you the importance of making the other person feel understood.
The technique basically boils down to repeating back your understanding of the situation to the person, and attempting to identify what they must be feeling.
If you're like me, this won't come naturally. It will feel fake, and you'll be worried that the other person will see right through you and accuse you of trying to therapize them. In my experience, this doesn't happen. If you keep the tone conversational and curious, the other person will be none the wiser, and will be happy that someone is finally trying to understand them.
An example might be if someone says to you "ugh, they keep giving me all the work to do while everyone else sits around". A non-empathetic response might be something like "maybe you could try delegating some of your work to the others", which isn't really an effort to understand the situation better, and more than likely will make the other person defensive.
To practice active listening, the conversation could instead go something like:
B: "Damn, that sounds frustrating. You must be pretty annoyed at the others for getting a free ride"
A: "No, I mean, it's not even their fault. I think I did this to myself by trying so hard to get my work done quickly when I started here".
B: "Oh, right. So they've identified you as the competent one, and now they feel like they can give you the most work to do since you can handle it?"
A: "That's right."
B: "Yeah, that sounds like a tough spot to be in. It's hard to go back to a lesser workload once people think you can handle a lot."
A: "Exactly."
What you're looking for is the person to say some version of "exactly" to confirm that you've got it right. If you can get to "exactly", you'll have successfully empathized with the other person.
Test it out one day. Make a deal with yourself to have an entire conversation where you don't suggest solutions, and only try to get a solid understanding of the other person's situation and their emotional state. You can always satisfy the solutionizing part of your brain by going on HN and sending some comments into the void.