In its simplest expression, flying is a balance between two sets of forces. Lift is generated by the flow of air over the wings. The red line is the leading edge, the air first hits the wing here. It travels up over the green line and then down the back of the wing where it leaves the wing at the blue line, the trailing edge. Basically, birds wings are not flat but are shaped like an aerofoil - concave.
Air passes over or under the wing as the bird moves forward, or as the wind blows.
The air that moves over the top of the wing has further to travel to get across the wing, thus it speeds up. This causes the pressure to drop because the same amount of air is exerting its pressure over a greater area. Therefore, any given point experiences less pressure. This effectively sucks the wing up. Meanwhile the air going below the wing experiences the opposite effect.