Empathy is the ability to imagine another person's thoughts, beliefs and feelings. In an ELT context, it normally refers to the ability to understand how a learner is feeling. It can be compared with sympathy, feeling sorry for how they are feeling.
A teacher working in a highly unstable and dangerous environment chooses not to do a reading comprehension where a writer describes their feelings after the violent death of their mother.
In the classroom:
Showing empathy with learners is important when they choose to communicate real feelings and experiences, for example in learner diaries or extended writing work. Similarly, it is important to understand the emotional difficulties connected with a learning experience.