How does the airflow around winglets reduce drag?
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.