Why Does the Air Speed Up?
When moving air encounters an obstacle—a person, a tree, a wing—its path narrows as it flows around the object. Even so, the amount of air moving past any point at any given moment within the airflow is the same. For this to happen, the air must either compress or speed up where its flow narrows. While air can be compressed more easily than water, freely flowing air acts much like water—at least at relatively low speeds. So when you "squeeze" a stream of air, two things happen. The air speeds up, and as it speeds up, its pressure—the force of the air pressing against the side of the object—goes down. When the air slows back down, its pressure goes back up.
Why does the air speed up? Because of conservation of mass, which states that mass is neither created nor destroyed, no matter what physical changes may take place. This means that if the area in which the air is moving narrows or widens, then the air has to speed up or slow down to maintain a constant amount of air moving through the area.
Why Does the Air Pressure Go Down?
For a stream of air to speed up, some of the energy from the random motion of the air molecules must be converted into the energy of forward stream flow. The random motion of air molecules is what causes air pressure; so transferring energy from the random motion to the stream flow results in lower air pressure.
Water Acts Like Air
You can see the Bernoulli principle at work in rivers. The water speeds up (and the pressure goes down) where the river narrows. The water slows down (and pressure goes up) where the river widens.
How Does All of This Create Lift?
A wing is shaped and tilted so the air moving over it moves faster than the air moving under it. As air speeds up, its pressure goes down. So the faster-moving air above exerts less pressure on the wing than the slower-moving air below. The result is an upward push on the wing—lift!