This illusion is known as Hermann's grid. The reason you see gray dots in the line crossings is because light-sensitive cells in retina of our eyes are arranged in rows. If a cell is stimulated an image is sent to the brain. However, if the adjacent cell is also stimulated the mind blurs the image. The effect is magnified and a high contrast black on white image.
Here is a slightly different version of Hermens Grid without the dots
