angrypickle
There are a great many different ways of explaining how wings create lift. As some of the above posts have revealed, the Bernoulli method is not universally accepted as being true. But you have asked the question:
"Why does increasing AoA (up to the crit AoA) increase CL?" Please keep it in relation to Bernoulli's theorem
So I will give you an answer to that specific question.
Bernouli’s theorem predicts that when the velocity of air increases, the static pressure will decrease.
When a wing is in flight the air stream meets it at a point known as the stagnation point. The air then splits into two streams with one flowing over the upper surface of the wing and the other flowing under the lower surface.
The speed of these two streams of air is determined by the length of the path which they take.
The length of this path is from the stagnation point to the trailing edge. If the lengths of the two paths are equal then the speeds and hence static pressure of the two streams will be equal. An example of this would be a symmetrical aerofoil at zero angle of attack. The upward force exerted by the pressure pushing up on the lower surface would be equal and opposite to the force exerted by the pressure pushing down on the upper surface. In this condition the two forces would cancel each other out, so there would be no resultant lift force and hence the Cl would be zero.
If we now pitch the wing 1 degree leading edge up, the stagnation point would move to a position slightly below the leading edge. This would increase the length of the path over the upper surface and decrease the length of the path under the lower surface. So the velocity of the air flowing over the upper surface would increase, thereby decreasing its static pressure, and the velocity of the stream flowing under the lower surface would decrease, thereby increasing the static pressure. The upward force exerted by the pressure on the lower surface would then be greater than the downward force exerted by the pressure on the upper surface, so the resultant force would be an upward (lift) force.
Gradually increasing the angle of attack further, increases this effect, thereby increasing Cl until the critical angle is reached, thereby causing the airflow over the upper surface to separate.
As I have said earlier in this post, the above arguments are not universally accepted, and it is very easy to pick holes in them. But I will leave that task to others who would prefer to show you how clever they are rather than attempting to help you.