It should be no mystery to the average high school student today just how airplanes get off the ground and stay that way. When we were kids we thought that it was the air pushing under the wings that made the airplane stay in the air and that by tilting the flaps down we would cause the airplane to lurch forward and downward and that by tilting the flaps up we would tilt the airplane up into the sky. Forward motion of course is a no- brainer - the engines provide thrust and push the plane forward and the wings would keep it in the air by all the reasons given above. Of course, we were usually talking about jet-fighters.
Well, so much for science. We didn't know then that the shape of the cross-section of the wing is the most crucial factor in 'giving the airplane lift', as they say. A wing has a flat surface on the bottom and a curved surface at the top. This curvature provides an overall greater surface area on the upper surface of the wing than on the bottom surface. The result is that since the same amount of air is passing over and under the wing, the air that passes over it is spread over a larger surface area and this creates a low pressure zone, or partial vacuum which actually causes the airplane to rise up. This is one manifestation of the Bernoulli principle as taught in high school physics.
Don't discount the role that the flaps play in flight, though. They play an important role in maneuverability and braking. In fact, a fighter jet traveling at a high rate of speed can indeed induce assisted lift by tilting the flaps up because the resistance that they offer to the air actually pushes the plane down at the rear and up at the front. This is why jet fighters are designed the way they are with the placement of the wing flaps toward the extreme rear to facilitate this process. Similarly, the airplane can be made to slow down more by tilting the flaps down as thrust is reduced. As the airplane's altitude decreases the air resistance increases because it becomes denser and the plane slows down even more. Tilting the flaps up on one side and down on the other will result in aerobatics, a spiraling motion.
In passenger planes, the flaps serve more to brake than to create additional lift although they facilitate this process as well. Any aeronautical engineer will tell you that every airplane manufactured has different technological specifications with respect to the functions of the wing and tail flaps, the weight distribution front to aft, the wing style and design and the flap placement not to mention the airplane's ultimate function, and all of these weigh into the design of the factors that allow an airplane to take off and fly. But again, the short answer to the question of how they do that is "the Bernoulli Principle."