I don't know about most animals, but I believe that dogs can read human emotions. I've experience it with my dogs. They seem to have a high capacity for empathy.
I think they can sense things that we don't pick up on. For instance, they can read our body language, the scents we put off out, the tone of our voice, the way we express ourselves. While they might not know our exact words, they are adept at interpreting those signs far better than we are.
I know when I was going through grief when my Dad died, my upsetting emotions upset my dog. I couldn't figure out why she had become so unruly. Then, it dawned on me that I was like a walking emotional time bomb and she was reacting to the energy that I was exuding. My emotional upheaval had a negative impact on her well being. So much so that she became a nervous, anxious, fearful dog. We were like two peas in a pod. Once I was aware of what was going on, I was able to help her through it. I don't know if she understood that it was grief, but she knew that something was not right with the pack and she reflected that.
I remember another incident involving my Lab. I was going through a terrible time with my arthritis and I was in pain a lot. She somehow knew my bad days. She would be very gentle around me. When she would come up on the bed to sit with me, she'd do it carefully. Then, she would lay her head on my leg very lightly as if she knew that I hurt. I always thought it amazing how she could read me.