How pilots and flight attendants are Paid is complex. Basic pay is from first movement of the aircraft to block in at the destination. To insure however you are not flown 1 leg and sat 5 days away from home there are also duty rigs. The industry norm is you get 1 hour of flight pay minimum for each 2 hours at work/on duty. In addition you get 1 hour of pay minimum for each 3.5 hours away from home. Some airlines also have an average daily minimum which at Delta is 5:15 per day you are gone.
In your case it sounds like a crew was put into rest to operate your flight later. Why that occurred I can’t tell you. If the crew arrived the day prior there should have been plenty of legal crew day to handle a long delay. It’s possible however your crew had already flown a leg that morning with a very early report. The FAA allowed duty day is based on time of report. If they reported at 4:45AM for a 5:45 departure to RDU their max duty day is 10 hours and they drop dead at 2:45 pm. They would need a 10 hour break to reset their duty period.