The author of that article (Graney) on Galileo's observations teaches at a nearby university, and I was able to attend a lecture he gave about the effect. The

The author of that article (Graney) on Galileo's observations teaches at a nearby university, and I was able to attend a lecture he gave about the effect. The tl;dr is that when you look through a telescope today stars are pinpoints. You can magnify them as much as you want, and they still appear pinpoints. Brighter stars might appear larger than smaller ones, but their size doesn't change with magnification. Galileo's telescope had such a small aperture that diffraction distorted the image, and the size of the disk of a star did change with magnification. Since they didn't understand diffraction at the time, they thought they were seeing the star's disk. Since there was no measurable parallax on stars at the time, they knew they had to be very far away, and thus unbelievable enormous in order to have a disk visible in a telescope.