Aerospace engineer here:
What a lot of people don't know is that the wings are actually installed on a small upward incline, relative to the longitudinal axis of the body. Think of holding your hand out the window of a moving car, and then tilting your hand to catch air under your palm. In aerospace we call this the Angle of Incidence, and most aircraft have a small amount, usually in the 1-5 degree range. So while you might be walking on a perfectly horizontal path as you go to the bathroom over the Atlantic on your way to Paris, the wings keeping you aloft are actually angled such that the leading edge is higher than the trailing edge by a small amount.
Now google any picture of an airfoil and notice that many of them are slightly concave on the underside. This is called Camber, and in a nutshell it creates a "cupping" effect under the wing that intensifies the high-pressure area under the wing and correspondingly increases the amount of air deflected downward. Additionally, the teardrop shape reduces the tendency of air to billow off the trailing edge of the wing in favour of kinda sticking to the wing's surface and following its curvature. This also causes downwash off the trailing edge (i.e. more air going downward, which is a good thing).
That's really all there is to it, from a high level. The wings deflect air downward such that the total momentum change causes an upward force that is exactly equal to the aircraft's weight, and that equilibrium of forces keeps the aircraft aloft.
Obviously it gets more complex than that, because guys spend entire PhD careers researching edge cases, but there's no magic involved.
Note that wings don't have to be of the classic teardrop shape. There are plenty of research papers about lift forces on flat plates. In fact that's classic fodder for an undergraduate assignment. The airfoil shape is beneficial in several ways, some of them quite subtle, but you can think of the airfoil as being the most efficient cross-section for a wing known to science, whereas a flat plate is much less efficient (though it still works).
>I even heard we don't completely understand why it works (?!?).
I don't think that's true. For a while there was the meme about "science says bumblebees shouldn't be able to fly" but that was a clickbait headline because we didn't know enough about the structure and motion of bumblebee wings. That's about all I can think of.
There are certainly areas of ongoing research and exploration (I'm thinking hypersonic flight, novel means of propulsion, aeroelastic structures, etc.) but in general, the physics behind conventional aircraft are quite well-understood.