Winglets are just low drag wing endplates that control the mixing of high pressure air from the lower surface, with low pressure air on the upper surface. By making it harder for these two flows to meet, you end up with an increased, effective, aerodynamic aspect ratio, which equates to about 1/3 the height of the winglet. Hence, the total wingspan effectively increases by 2/3 the height of one winglet.
By controlling the spill over from lower to upper, the outer wing generates more lift resulting in an increase of CLmax. The CLmax increase shows up in reduced stall speed, which leads to reduced takeoff and landing speeds and distances.
Winglets work best in the high CL environment, e.g., takeoff, climb, high altitude cruise, and landing.
Most winglets on GA are for looks.