The study, conducted by researches from Columbia University, NASA and the Logistics Management Institute, constructed models based on the five most popular aircraft and nineteen major airports around the world. The study says, "We find that on average, 10–30% of annual flights departing at the time of daily maximum temperature may require some weight restriction below their maximum takeoff weights, with mean restrictions ranging from 0.5 to 4% of total aircraft payload and fuel capacity by mid- to late century."
The problem is in the density of the air: When air warms, it becomes less dense and more expansive than colder air. This means that in order to take off in warmer temperatures, an airplane needs to be at a higher speed than if, all else equal, it were taking off in cooler temperatures. As air temperatures rise, flights will need to have longer runways or weigh less in order to take off.
The researchers note that they tested both small and large aircraft in their models, and both were affected by higher temperatures. There are going to be some real repercussions for this; airlines are going to cancel and reschedule flights more due to excessive heat, and airlines will have to start investing in infrastructure to extend runways. Additionally, less weight on an airline means fewer passengers and baggage, so you can count on ticket prices and check-in bag fees increasing in the medium-term futures.