As you can see from the illustration, the flapped wing has both a higher coefficient of lift (CL) and a lower angle of attack at the stall (the peak). Since the flaps are inboard, that section of wing will stall first when they are in the extended position.

Next time you’re out doing pattern work, do a no-flaps landing and a full-flaps landing. If you shoot for having the stall warning going off constantly at touchdown (light or no-wind conditions), you’ll notice how high the nose is without flaps and how low it is with full flaps (even if you’re not really stalled). You’ll also notice that there’s less time for the flare with full flaps, due to the steeper descent angle and much higher drag/faster deceleration.

The stall during the no-flaps landing is more “equal” along the length of the wing (not counting what the manufacturer has built in to ensure that the inboard section truly does stall first).

The stall with full flaps is a very different situation. The outboard, unflapped section of wing is nowhere near the high angle of attack it had on the no-flaps landing. The stall and much lower nose position is being driven by the flapped inboard section. The outboard section of wing is at a much lower CL now. It’s not adding much in the lift department, but is still giving that great aileron control we all know and love when near the stall. Realizing the big difference in lift between the parts of the wings, you can see why some short-field airplanes have ailerons that droop a bit as the flaps extend: to add some lift and drag out there, but still stall last.
Even though the flaps generally only affect part of the wing, we know that there’s a “net” increase in coefficient of lift with flaps (and a big increase in coefficient of drag) by the fact that the stall (in calibrated airspeed) is reduced by using flaps. Here are the the numbers at gross weight with 0° of bank for a certain (ahem) airplane:

Flaps up: 56 KCAS

Flaps 20°: 51 KCAS

Flaps 40° (full): 50 KCAS

You can see that there’s a big reduction in stall speed by using 20° of flaps (9%). The use of full flaps reduces it a little more (2%), but we know that it increases the drag bigly.

Landing on a normal runway with full flaps won’t shorten the landing roll much, since it will only reduce kinetic energy by about 4% from Flaps 20, but will certainly require a much steeper approach or a lot more power and make the go-around (that we should always be ready for) more complicated.

Here’s an excerpt from the newest (9th) edition of The Advanced Pilot’s Flight Manual:

“In choosing a landing flap setting for everyday flying (not short/soft field or downwind with limited runway), consider the flap setting the POH calls for on the go-around (or less). Using this intermediate setting will give good landing performance, yet without requiring you to handle (and possibly mis-handle) the flap lever close to the ground when you’re very busy controlling the airplane (low airspeed/high power/high torque).”