The vortex spins because of centripetal force - the one that famously makes an iceskater spinning on the spot twirl faster when his or her arms are pulled into their body. As the water spins, it moves faster at the base of the bottle, where the bottle neck is smaller. The faster-moving water pulls the water down into the bottle.
At the same time, air from the bottom bottle is let into the top bottle because of the shape of the vortex, which allows a funnel for the air to flow. Because the air and water can both flow freely while the water is spinning, the water pours into the bottom bottle faster than it did when there was no vortex.
