Lift can be described as a moving wing colliding with air molecules at an angle, the result of the collision is the wing moves one way and the air mass the other, as per momentum physics.
Moving the trailing edge, or the entire surface, increases the angle of attack, resulting in more lift at a given speed $V$:
$Lift$ = 1/2 × Lift Coefficient x Air Density x $V^2$ x Wing Area
Deflection of the control surface produces a linear response to lift, as Peter Kampf says.
A graph of angle of attack vs lift coefficient generally shows a linear relationship through most of the unstalled AOA range.
Further inspection of the effect on AOA, of deflecting a trailing edge flap down, as compared with "drooping" a leading edge, does indeed show doing this changes the angle of attack of the lifting surface!
Flaps are usually near the wing roots, and leading edge slats near the wing tips, for this reason. We want the root to stall first.
Yes, deflecting a control surface changes the camber of the wing, which also factors into the lift coefficient, but, relative to the original AOA of the wing/fuselage, the flapped portion of the wing will stall at a lower AOA than the slatted portion. Therefor, deflecting the control surface also changes AOA.
The importance of this concept is highlighted in a slow flight coordinated turn. Use of ailerons without coordinating rudder may result in a sharp roll in the opposite direction of the intended roll, because the AOA of the "down" aileron wing now exceeds its stall AOA.