How does the airflow around winglets reduce drag? I am familiar with the basic aerodynamics of wingtip vortex generation and I know that winglets reduce drag by reducing wingtip vortex generation.  I have never seen any illustration of exactly how winglets interact with wingtip vortices to do this.  I have also heard there is a small thrust component to the winglet's lift but am also unable to visualize this as well.  The multiple designs in use are confusing.  Why not just curve the end of the wing down to inhibit the vortex from spilling upwards?
This video perfectly explains what happens, except it doesn't use winglets, but different shape of the wings, but the idea is the same:

https://www.youtube.com/watch?v=CnvIf3vFEYA

What the video describes is when wings don't have winglets, air flowing under the tip of the wing escapes to the side of the wing and upwards, creating a vortex which reduces lift. Addition of winglet reduces the vortex and thus increases lift. 

In the video however, new type of wing is shows that doesn't require winglets all together.