There's denser details inside and sparser details further away, creating a focus point, which is generally considered a pleasing composition... Besides that, the examples don't really show anything that useful or magic as far as I can see. (A negligible part of an arbitrary portion of Sonic sort of lines up to the curve? One box is the same height as monalisa's head, and one other box is sort of aligned to some of her nose, but only if you scale and offset the grid that way?) Interesting experiment, but if God's fingerprints are on your Sonic sprite, I don't think it's really benefiting from them.
I have seen a couple more convincing applications than these examples, but it's just a tool, likely often unintentional and holds no real mystique IMHO. The Apple logo is pretty interesting, but I guess it's more to do with easy recreation in the pre vector-software world? (I'm not sure, but it makes sense so you can scale the whole thing by one measurement).