Back to donuts for a second. Here are some crude images.
Without a donut light from anywhere on the lens can and does hit anywhere on stage. So you have stray light that doesn't hit where it is "supposed" to. This is what yields unsharp images.
With a donut you cut down on the extraneous light and only allow the light to go where it is supposed to go. Consider that any point on an object emits light in all directions, but to get a sharp image you only want the light ray that comes from that point, through any optics that hits the same point in the image on the stage. The ray that comes out at a slightly different angle will focus through the optics differently and not hit the same point on the image on stage, thus making it not as sharp.
Without a donut light from anywhere on the lens can and does hit anywhere on stage. So you have stray light that doesn't hit where it is "supposed" to. This is what yields unsharp images.
With a donut you cut down on the extraneous light and only allow the light to go where it is supposed to go. Consider that any point on an object emits light in all directions, but to get a sharp image you only want the light ray that comes from that point, through any optics that hits the same point in the image on the stage. The ray that comes out at a slightly different angle will focus through the optics differently and not hit the same point on the image on stage, thus making it not as sharp.