Credit card fraud is a significant issue for airlines. See, for instance, this 2012 article:
Some of the travel agencies in the criminal underground are
full-service, pitching package deals that include airfare, car
rentals and even hotel stays. A hacker using the nickname “Yoshimo” on
one prominent fraudster forum offers “80-95 percent working flight
tickets in most countries (some restrictions apply),” for 25 percent
of the original price, and 40 percent of the price for carded hotel
stays and car rentals. He has been offering this service for more than
two years, and has at least 275 positive reviews from current and
At first glance, it may seem unlikely that your typical paranoid
fraudster would dare take advantage of such a service. But according
to the proprietors, few customers are ever stopped, and those that are
can simply claim that they were victims of fraud. At least that’s how
it’s explained by “Jeferi,” a criminal travel agent who has set up
shop on the fraudster forum Kurupt.su.
As search on "airline credit card fraud" turns up many other stories on the situation. Here's 140 people detained and questioned over two days around the world over plane tickets purchased with stolen credit cards.
Such fraud is difficult to fight, because passengers can claim, often reasonably, that they are victims themselves and are unaware their tickets were purchased with a stolen card. You don't know, if you buy your tickets from a dodgy website or travel agent offering a low fare, whether they really pocketed your money and gave you a stolen ticket. Actually finding the person responsible for the fraudulent charge is far more difficult.
As such, airlines may require you to show the credit card that was used to purchase the ticket before allowing you to check-in. This doesn't eliminate all fraud, but it does stop those who are unwittingly flying on stolen tickets, as they will not have the correct card in their possession. In the past, such requirements used to be applied to many more passengers, but airlines recognize that buying tickets for employees (though the rule may be waived for flights under corporate contracts or booked through corporate travel agencies), friends, and family is common, so they've relaxed the policies and allow many passengers to fly without this verification.
However, in situations where the airline sees a greater risk of fraud, perhaps based on their experience, they impose greater verification requirements. This is especially true for low-cost-carriers, as they are less likely to eat the cost of some fraud in the name of improved customer relations by not turning legitimate passengers away. In the case of flydubai, their policy states that they may always require to see the card, but they will accept "a photocopy of the front of your credit/debit card and a photocopy of your passport" unless you are traveling to/from one of the listed countries. For those countries, they have apparently decided that the risk of fraud is too great.
It's important to carefully check the airline's policy (or ask a question here if you're not sure) before buying a ticket for someone else with your credit card.