I've had to file a claim twice. First time was coming home from a cruise and we got stuck in Cincinnati. They closed the airport due to tornadoes. Second time was on the Conquest in November 2006 when someone jumped off the ship the last night out.
First, I got all the information available about why there was a delay. For the airport being closed, I got a statement from the airline. For the Conquest, I printed news articles about the incident.
Second, I called the insurance company as soon as I could, which was before disembarking the ship and after the airport closed and we knew we would have to spend the night.
Third, I saved all receipts for the cost to change flights, hotel room, rental car, lost luggage claims and food, etc.
Fourth, I made sure all my paperwork was completed correctly and kept copies of everything I had sent.
So far, no problems receiving reimbursement.