We are very accustomed to gridded views. Not only looking through the gridded windows, but when we walk along the streets, blocks of houses, shops by the sides, with light or darkness coming through them, build our respective impression strong through the absorbed memories of grid views. Neglecting the curvature, the earth's surface is horizontal, and on it gravitational force is perpendicular. For that, our world is more horizontal and vertical in nature than any other form. And this is the reason behind Hermann-Grid optical illusion.
In Hermann-Grid illusion, we challenge 'the possibility' and 'the impossibility'. If the inside light is so bright to make the grid complete white, the darkness beyond the grid could not be as dark as complete black due to reflection and scattering of light by the grid. When we look at a bright light source, there appears a hallow around it due to scattering of light by the atmospheric molecules and particles. The darkness around the light source is smudged with same tinge of colour. To watch background as dark as complete black, the insight should be less lighted or the grid should be darkish. So the illusion appears at the peripheral vision which lies between the possibility and the impossibility. In Lingelbach illusion the borders have already been smudged with darkness. So the illusion does not appear at the borders as we are accustomed to see that type of darkness only through that colored grid.
In the vice versa case, if the background is bright white, bright yellow, bright red or of any bright colour, and the grid is of different colour, the illusion would also appear, because due to scattering of light, and to some extent reflection of the light from the inside objects including the observer, the grid will have a tinge of that colour. So for theoretical purpose, for easier understanding, whether the background is dark or bright we can presume that background is scattering its own colour to produce its effect on the grid.
Now, why the darkness or smudge (effect) is more prominent at the intersections? – It is because of overlapping areas of scattering halos (from four quarters) at the intersections.
Why we do not see the illusion when the grid is tilted at 45 degree or the shape of the grid is changed?
Because we are not accustomed to see those views. We don't see any more those as usual familiar views; we just see those as a design or a painting that does not challenge our memory.