Why and how do flaps increase the curvature of the wing?
Flaps change both the camber of a wing as well as its chord line and angle of attack. All the following images were taken from this article.

You can see the camber line here in a retracted scenario:



And here in an extended scenario:



As for the why, flaps are generally used to allow you to slow an aircraft down at a give pitch angle. This allows the pilot to pitch the aircraft down without an increase in speed. Generally a pitch down (given no other things are changed) will result in an increase in speed. 

Flaps also change the angle of attack of the wing as well as the chord line:





All of this is a bit of an over generalization as there are many types of flaps and it's worth looking into the different types and how they operate.