I'm sorry you feel that way about it
It's a phrase I've heard many times, usually in response to a complaint. I always take it to mean that the person being told the bad news, whilst acknowledging what's been said, is being careful not to agree with it.
Sometimes a complaint is completely unjustified, and the person raising an issue is simply venting their emotions because they are upset about their own, or their loved ones, illness. In those instances, they need to be dealt with by someone with the sensitivity to reduce their stress, someone who has an intuitive understanding of their underlying anxieity. On the other hand, if that person has a geniuine grievance, someone should be big enough to apologise.
Imagine if your child had just been naughty. How would you feel if they responded to their misdemeanor by saying 'I'm sorry you feel that way about it'. Surely, it would make you even more angry.
So often, simply saying your are genuinely sorry is enough to placate someone. Failing to acknowledge someone's unhappiness with genuine understanding, especially if that someone is a patient or relative who is already suffering in other ways, can be hugely insensitive.
It's called compassion.