I hope that helps some, this is intentionally not a rigorous answer, and I want to recognize that I am not addressing the more hairy details of the actual fluid equations associated with this, which are required for a full explanation.
The curved top, however, increases efficiency by intensifying that natural effect. A plane can fly upside down, but I don't know of a plane that can maintain altitude with the wings not angled up. A simple argument for this is that the point of separation is lower than the front of the wing, again, since the wing is angled up.
Such a mode of flying, however, will still see the fluid passing over the top of the wing faster. A plane can function with no additional curve on the top of the wing, as the famous xkcd comic points out. Now, you may say, "but it will flow faster over the top even if the top isn't curved more!" You would be correct. If both paths take roughly the same time to pass over the wing, then the average velocity of the fluid from the point of separation to the tail where the flow rejoins will be roughly proportional to the distance from those two points. First identify the point at which the flow separates, meaning the point above which the fluid goes over and below which the fluid goes under, this is slightly below the front-most point of the airfoil, due to the fact that it's angled slightly upward. The argument that the wind flows faster over the top is mostly a consequence of geometry. Looking at Wikipedia, I'll post two images:
To start with, I think we need to identify the point at which the flow separates. However, I understand why you would find this explanation unsatisfactory. The common explanation given is that it flows faster over the top of the wing because the top is more curved than the bottom of the wing. When the air hits the front of the wing it flows in a steeper curve upward, than the bottom wing flow, This creates a vacuum on top of the wing, and this pulls more air towards the top of the wing, this air does the same thing but moves faster because of the vacuum pulling it in, and then the vacuum of course lifts the wing.