Uplinker - There is a simple way to understand the main advantages of winglets...
High aspect ratio wings (eg long span narrow chord) are more efficient than low aspect ratio wings (short span, wide chord). Thats because the disturbed air at the wing tip represents less of a fraction of the span if the span is bigger. So anything you can do to increase the aspect ratio is generally a good thing.
Unfortunately you can't just increase the wing span indefinitely for practical real world reasons. Winglets work by increase the effective aspect ratio without increasing the span.
They aren't a totally free lunch. Increasing the efficiency at the tip effectively moves the center of lift (of each wing) outboard increasing the bending moment.
The AOA of the wing is easy to control. It's not so easy to adjust the AOA of the winglet. So the winglet might have to be optimised for one particular speed (eg cruise).