Explanation

When you empty a bottle the water leaves, but air has to take its place, so in the narrow neck of the bottle the air has to pass around the water. If you just turn the bottle upside down this happens in a series of 'glugs' as some water falls out, then a bubble of air passes up the neck of the bottle then another lump of water falls out. This means that the water has to speed up then slow down every 'glug' which is a very slow process.

The spinning water leaves a path up the centre for the air to enter

For the water to leave the bottle it has to let air in past it The spinning water leaves a path for the air to enter.

If you spin the liquid it is thrown out to the sides of the bottle, leaving a gap up the middle which the air can travel. So although the path of the water is much longer it will still empty faster.

Of course an even faster way of emptying the bottle would be to drill a hole in the base of the bottle so the air could get in directly, though that would be cheating.