I think that they can sense our emotions, and want to support us when we're upset. I don't think you need to be concerned about it being too stressful for them though. If you're worried about, I'm sure they'd appreciate some extra cuddle time with you.
This isn't directly related, but I also believe that cats can sense illness. When I still lived at home & before my father was diagnosed with cancer, he had extreme back pain but would not go to the doctor. One night, I was up with him because he couldn't sleep due to the discomfort. Both the family cat & dog were up and following him around EVERYWHERE he went, which was unusual for both of them, but especially the cat who tended to keep her distance. That night he ended up going to the emergency room because the pain was so unbearable; he was diagnosed with cancer soon after. I think they knew, and I will never forget that experience. I believe our furry friends are very intuitive.