One proven way to improve one's empathy skills is to simply Get A Pet, like a dog or a cat.
Since animals don't have the ability to communicate with us humans on a verbal level, understanding what they really feel will be hard. One needs to rely on their gut feeling and will need a little rational thinking in taking good care of their pet.
And this is where empathy comes in. Over time, pet owners will start to understand the little actions and the small behavioural changes their pets manifested. They will then act in accordance to this small changes as slowly by slowly they are learning what empathy is.
Since animals can't talk, dealing with humans will be much more easier. For one, they are more vocal with their feelings, others also can't hide their facial expressions and the tone in their voice, one that animals can't do..