Too many thoughts
Here whe have to define Angle of Atack precisely.
I think that the typical graphs showing how stalling AoA decreases when lowering trailing edge flaps are referred to the "airplane AoA", that is, to the angle of the wind relative to the airplane's longitudinal axis.
Strictly speaking, an airfoil's AoA is the angle of the wind relative to the it's chord. When lowering trailing edge flaps you actually change the airfoil. And these airfoils have different dimensions, characteristics and performance.
Their chords are different, so: How to compare stalling AoAs?
In the graph we see flaps down curve parallel and above clean curve. But why the stall occurs at lower (airplane's) AoA?
And another question (the more I think about it, the more questions, Argh!):
Why is the flaps curve parallel??
The reason for the "anticipated" stall when flaps are down is, I think, the increased airflow circulation due to the increased camber. The negative pressure gradient after the maximum camber is more intense, so separation occurs earlier. I suppose the answer to the 1st question is this fact plus the fact that flap airfoil's chord angle of incidence is increased with respect to clean airfoil, so everything has to occur earlier with respect to the airplane's logitudinal axis.
On the other hand, flaps down zero Lift occurs at an earlier AoA, too. According to the typical graphs, the flaps down range of possible positive Lift AoAs is much greater than the decrease in stall AoA. I suppose this can be read as "camber increases lifting capability of an airfoil". But still I don't understand why the slope of both curves is exactly the same. Or are they not?