First of all as nurses we need to realize our patients are often times reacting out of fears and personal concerns such as cost of care received, being away from home and love ones among other concerns. Active listening and showing empathy and compassion goes a long way with further communication with a difficult patient ( or family member). Humans need to feel safe and need to know they are heard. I have found by taking the extra time initially I am able to complete my duties easier while also lessening extra time needed to handle difficult patients. Many times it means checking on them a little more frequently to show you really do care.
That's a standard behavioral interview question in any interview. (Take a moment and google "behavioral interview questions.") I would interpret the question as, "Tell me a story about a situation in which you were successfully able to resolve a near-impossible problem involving a severely angry or upset patient." The object isn't to tell me what a good nurse you are. It's to tell me how YOU were able to solve a problem that most nurses would not have been capable or willing to resolve. This is where you show off your skills in analysis, communication, and storytelling. A good answer would involve a patient with a problem that has been repeatedly mishandled because the staff didn't have time to listen and or take steps to resolve the situation. At first, you misunderstood as well, but then you looked deeper and discovered the problem was much more serious than it seemed. You got on the phone and called the right people to get things moving, but there were problems still. You stayed on it until everything was resolved and the patient lived happily ever after. Then, after saving the day, you took measures to ensure that never happened to another patient in that institution.
The important thing is that you are differentiating yourself from all the other nurses the interviewer has seen today by showing off your keen insight and good choices under pressure. You're also recognizing the interviewer is bored and wants you to pull them in with a good story. If you've worked as a nurse for a while, you'll have a long list of these stories but it's easy to go blank if you're not expecting the question...
Look up trauma informed care and motivational interviewing. These are compassionate tools that can be very helpful in these situations that will pop up again and again. It's up to you to be professional, and these are good tactics for deescalization.