This is only partially true, though; a totally flat wing can also support flight. The shaped nature of the wing contributes to its efficiency (and other factors) but do not make other wing shapes incapable of supporting flight.
The reality is that the Wright brothers' innovation was not the airfoil shape or even the lightweight motor. It was the control surfaces, to allow the operator to adjust the plane's attitude on the three axes of rotation, allowing actively stabilized flight.
Paper airplanes and kites demonstrate all the same principles of heavier-than-air flight (the Wright brothers even had a kite version of their airframe they used for testing), despite the fact that they generally do not exhibit shaped airfoils.
Without those controls, flight is basically impossible, and with them, you could use nearly any airfoil shape (modulo engine power, drag, and stall speed considerations) and achieve heavier-than-air flight.
For example, yes, the air above the wing moves faster than the air below the wing, and it's related to shape of the airfoil.
However, it has nothing to do with magical "air has longer to travel".
It starts with how combining flows at the trailing edge of the airflow create a vortex which induces an opposite vortex around the wing, which is a bit counter-intuitive (but it has nothing on why swept wings work, which can be summarised for practical aircraft design purposes of "because if we calculate at an angle we get better values and reality is crying in the corner")
Wait, I was under the impression this Cutta circulation was a computational simplification and the "real" reason were the pressure differences as explained in this submission. What am I missing?
One comment already mentioned how position of flaps could have visible effect on pressure sensors in front of the plane, and this is slightly mentioned in how the pressure created by front of the air foil has an impact on air "at a distance" from the airfoil.
The vortices created around the airfoil result in significant change of flows, which especially at low speeds provides big chunk of the pressure changes necessary for the creation of lift, with the effect IIRC getting lower as you go faster, with transsonic regime breaking it - because that's when the resulting speeds go beyond speed of sound at given pressure in the air, which in very simplified way means that air can't move towards front of aircraft anymore in those areas, breaking all sorts of flows you depend on at lower speeds.
It's the size of the aerodynamic forces and the complexity of the physical mechanisms that create them that many people have trouble with. In particular: intuitions can be pretty wrong, most simplified explanations are wrong under simple experiments, and the problems exhibit scale variance that is unfamiliar (e.g. Reynolds number).
One time I was working on air data computer for a transonic aircraft that could fly up to about M0.95 - during flight test, an air data probe mounted on a nose boom was used to supply impact and static pressures, angle of attack and sideslip etc. for various air data calculations like airspeed and altitude.
I was fascinated that there was a term in the calculation that related to the aircraft flap position - what's happening way out on the trailing edge of the wing actually has a meaningful effect of pressures measured on a boom out the front of the nose during certain regimes of flight.
I mean, actually, it isn't - that's the whole point about scale variance and Reynolds number and why wings that work for insects are not the wings that work for jumbo jets.
We still don't have a very good understanding of turbulence.
Navier Stokes equations still make Aerospace engineers drink.
Yes its stupid simple if you care about the simplest of analogies, but if you try to understand it, there are reasons why 80% of people drop out.
https://www.youtube.com/watch?v=NBsvzMi9-f8
So yeah, fans are puzzling too.
I'm guessing the initial puff creates a high pressure area on top of the paper, rolling it downward and back. Them after the puff has pushed the air away, there is now a low pressure zone on top of the paper which lifts it up as the air below is rushing upwards around the sides of the paper.