I suppose what I said is a little misleading, in the sense that more lift is not created by (as in your example) changing the geometry of the wings. In your case of extending the flaps, additional lift is generated by increasing the wing area, lengthing the chord, and putting more camber in the wing section. Of course, as you mentioned, the penalty you pay is an increase in parasitic and induced drag, as well as pitching moment.
Now here is where it gets a little involved without pictures. The tip vortices produce a local upwash in front of the wing and a downwash behind the wing. The practical effect of this is to reduce the "local" angle of attack (near the wing as opposed far away where the upwash has no effect) that the wing sees and thereby reducing the lift.
Here's the tricky part: lift is defined as a force perpendicular to the direction of the airflow and drag as being parallel to it. The difference between the lift defined by the freestream direction and the lift defined by the "local" direction ends up producing a force in the freestream drag direction. Hence the name: induced drag.
The upshot of all this is that if you reduce the induced drag, you recover the lift that was lost, i.e. more lift.
P.S. Just as a side note, none of this impresses the little ladies. I'd stick with the daring exploits and embellished war stories.
LOL - Just read your profile. You probably already knew all that.