I'm wondering if the OP wasn't after a much simpler answer...
On one level you can consider the lift produced by a wing to be more or less constant and equal to the weight of the aircraft even when climbing.
Increasing the angle of attack allows a wing to produce that amount of lift at slower speed but only within limits due to the stall.
Deploying flaps allows the wing to produce that amount of lift at an even slower speed to make landing and takeoff safer.
The extra drag produced by the flaps is worth it.