How do you emotionally support someone?

I struggle with this. A lot. I know that no one is the same and that no one solution will work every time. However, I don't think I really have foundational knowledge on how to do this--generalized skills that are pretty versatile that can be adapted to cover most situations.

This is negatively impacting my relationship with my partner a lot, and they're at their breaking point. It tears me up inside seeing that I struggle so much and cause them harm. But I want to change. I want to have confidence in my efforts. I feel very stressed and anxious about this.

I don't know what more to do. They're willing to answer questions, but don't have the mental bandwidth to explain support to me.

I want to learn and quickly. Please, help.