Air flow follows pressure gradients, and the lift-creating suction over the upper wing pulls in more air, not only from ahead of the airplane (which explains the induced angle of attack), but also from the side.
The explanation of induced drag on the Boldmethod page is actually quite good; only gems like the wingtip vortices curve up and around the wingtips, pushing the air flowing over the wing downward and the graph with straight arrows on a swept wing are in need of improvement.
The better wording would be: Lift is created by accelerating the air flowing over the wing downward; this in turn causes more air from the side to be accelerated towards the center of the wing.
Regarding the comparison of winglets to sails: The sideways acceleration of the air near the tip causes an inward-bent airflow above the wingtip. If you now place a sideforce-creating surface there, its lift vector will be tilted forward by this inward-bent airflow. This will produce a small thrust component together with a large side force, just like a sail does when sailing close to the wind.
This effect works in the same way as the downward-bent flow over the wing which causes the wing's lift vector to be bent slightly backwards, causing induced drag. You might, therefore, call the beneficial effect of the winglet induced thrust. Just like induced drag, it is biggest at high lift coefficients.
Of course, folding this winglet flat out as a span extension will be even more beneficial.