This isn't a question of how a wing works — vortex flow, Bernoulli's principle, all of that jazz. Instead, it's a question of why we need a wing at all. A wing produces lift, but why is that necessary?
I got to this by thinking of an airplane at a coarse level. The wing produces lift through some interesting physics, but it needs energy to do this. The engine is what ultimately provides all of this energy (let's assume no headwind, and in "ultimately" I'm not including chemical energy in the fuel, yadda yadda "it all comes from the sun"). That means the engine pushes enough air, and fast enough, to (a) offset gravity and (b) still propel the plane forward. So the question is: why can't we just angle the engine down a bit and get the same effect?
To slightly reword: why do wings help us divert part of an engine's energy downward in a way that's more efficient than just angling the engine?
One answer is that we can do exactly that; I'm guessing it's what helicopters and VTOL airplanes like the Harrier do. But that's less efficient. Why?
One analogy that comes to mind is that of a car moving uphill. The engine doesn't have the strength to do it alone, so we use gears; for every ~2.5 rotations the engine makes, the wheel makes one, stronger rotation. This makes intuitive sense to me: in layman's terms, the gears convert some of the engine's speed-energy into strength-energy.
Is this analogy applicable — is the wing on a plane like the gearbox in my transmission? And if so, what's the wing doing, more concretely? If a gear converts angular speed to increased force, what X does a wing convert to what Y?
None of the answers I could guess at satisfied my intuition. If the wing converts horizontal speed to vertical speed, tipping the engine downward would seem to have the same effect. If it's changing the volume/speed of the air (more air blown slower, or less air blown faster), it would still have to obey the conservation of energy, meaning that the total amount of kinetic energy of the air is the same — again suggesting that the engine could just be tipped down.
EDIT
In thinking about this more from the answers provided, I've narrowed down my question. Let's say we want a certain amount of forward force $S$ (to combat friction and maintain speed) and a certain amount of lift $L$ (to combat gravity and maintain altitude). If we tilt our engine, the forces required look like this:
The total amount of force required is $F = \sqrt{S^2 + L^2}$. That seems pretty efficient to me; how can a horizontal engine + wing produce the same $S$ and $L$ with a smaller $F'$?
Best Answer
Let's look at the relationship between momentum and energy. As you know, for a mass $m$ kinetic energy is $\frac12mv^2$ and momentum is $mv$ - in other words energy is $\frac{p^2}{2m}$
Now to counter the force of gravity we need to transfer momentum to the air: $F\Delta t = \Delta(mv)$
The same momentum can be achieved with a large mass, low velocity as with small mass, high velocity. But while the momentum of these two is the same, THE ENERGY IS NOT.
And therein lies the rub. A large wing can "move a lot of air a little bit" - meaning less kinetic energy is imparted to the air. This means it is a more efficient way to stay in the air.
This is also the reason that long thin wings are more efficient: they "lightly touch a lot of air", moving none of it very much.
Trying to replicate this efficiency with an engine is very hard: you need compressors for it to work at all (so you can mix air with fuel and have the thrust come out the back) and this means you will have a small volume of high velocity gas to develop thrust. That means a lot of energy is carried away by the gas. Think about the noise of an engine - that's mostly that high velocity gas. Now think of a glider: why is it so silent? Because a lot of air moves very gently.
I tried to stay away from the math but hope the principle is clear from this.