Simultaneous contrast
This illusion is similar to the Grey Steps Illusion. These illusions work because of the way your eyes see colours and edges.
When the tail covers the division between the black and white parts of the board, one half of the grey circle appears darker than the other. When the division is not covered, the circle appears to be the same shade of grey all the way around. This effect is known as simultaneous contrast. The shade of an object appears to change, depending on whether it is seen against a light or a dark background.
Why does it happen?
There are a number of explanations for why this happens. All these explanations involve the complex communication system that links the eyes to the brain. When we look at the horsetail illusion, we are comparing the grey circle with the area that surrounds it. When the horse's tail is placed down the middle, each half is compared separately to the area surrounding it. But when you take the horse's tail away, the grey circle is seen as a whole, so our brain knows it must be all one colour.
The brain looks for differences between parts of what we see in order to decide what is there. When it finds differences, it exaggerates, to make it easier to tell things apart. Sometimes this means things that are actually the same colour can look different.
You have probably noticed that when you wear different combinations of clothes, some items seem to change colour slightly, depending on what you wear with them. For example, when you see yellow against a green background it looks more orange-red than the same yellow on a white background.