Barrel Flying is a tactic possible only in singleplayer. To do it, you have to stand on a barrel, then you have to put your cursor on the barrel and then to activate the Kinetic Module (press and hold the C button). Now, slowly lift the barrel to any spot you want. This will make the barrel fly with you on the top of it. It's recommended to take care to avoid sharp turns as you could fly off the barrel, and try not to get shot out of mid-air.
This technique is usually used to get on top of high walls or to fly high over any enemies to the end of the level.