How do endplates reduce induced drag?
The vortex itself isn't the issue, it's the "leakage" that is resulting in the vortex.  The leakage is from the air near the tip being free to flow around it and results in wasted energy.  

The wing is moving along driving air down, but near the tip some of the air, instead of being made to accelerate down, is taking the path of least resistance around the tip (being sucked toward the low pressure side you might say) instead and doesn't contribute to the lift force.  It was displaced, and energy was consumed, but the displacement was out and around the tip instead of down.  

The tip plate is just supposed to inhibit some part of the lateral flow around the tip by blocking it, reducing the leakage and resulting in more of the air package being forced downward than without the tip plate.  What is referred to as a reduction in induced drag is just an improvement of the ratio of lift to drag due to less leakage losses, so for a given amount of total lift force, the induced drag is less than for the same lift force without the tip plates.

To do any good, tip plates have to be quite large (because the circulation field extends quite a distance from the wing), which being draggy themselves, cancels out most of the benefit, which is why you don't really see them used very much, except where the end plate effect can be created by default, such as with a T tail where the horizontal stab makes a nice end plate for the top end of the rudder.