Lyman...
Are you familiar with F=ma? If you are, you'll know that for a mass (ie: a book, car, computer, apple or air) to be accelerated, there must be a net force. A net force means an 'excess' or imbalance in the forces on the object. For example: When a book is resting on the table, there is no net force on the book, therefore the book is not accelerating. Since the start velocity was zero and there is no acceleration, the book will continue to lie on the table forever... unless there is a net force applied to it.
Pressure is essentially a type of force - think of it as the source of a force. This could be water pressure (force) from a firehose that knocks you off your feet. So for the air to be accelerated, there must be a net force applied to it. Just as a clarification, an acceleration is required to change either direction OR speed since acceleration is related to velocity and velocity is a vector quantity (meaning it has both a magnitude (speed) and direction). So in the case of the air flowing over the wing, it is the low pressure (force) that is applied to the air that accelerates it. If the air runs into a higher pressure area (like as it passes underneath the wing, close to the surface) it will experience a net force and accelerate in a negative direction (aka: slow down).
In all cases here we're dealing with static pressure and not dynamic pressure so the actual speed of the air does not change the pressure measured - don't think Bernoulli here! For example: most test airplanes will have a very long probe attached to the nose of the airplane to measure free stream attributes - this is before the air is affected by the airplane. It doesn't matter how fast the airplane flies, the static pressure measured will be the same as long as the airplane is flown at a constant pressure level such as FL050.
The air is acceleratd by the higher pressure
Professor Nicholson demanded that in Nature, there was no "pull" only push.
To me, all that is is relativity. High pressure pushes, low pressure pulls is the same thing. It's a property of fluids that they try to equalize so that there is no net internal force applied by itself. The reason there is higher pressure at sea level and lower pressure as you climb is essentially just because there is gravity.