Keep it simple. Think through the steps you take when handling demanding customers. Be prepared to give an example of how you have dealt with these types of customers in the past. If it's all new to you, be honest! Explain how you would use your communication and listening skills to relate to customers and calm them down. And tell them you're learning! You aren't expected to know everything in a new job, so just relax and do your best!
"I stay calm, and I listen. I'll let a customer vent as long as they need to. Then I empathize and look for a solution or a compromise."