"Our shapes cast shadows in their minds"
The Nazgûl's minds. 'They do not perceive the world of light as we do', says Aragorn.
Gandalf says a wraith 'walks in the twilight'.
Light here is clearly sunlight. We walk under the sun and become 'weary' under the sun, to use the word Faramir uses when talking to Frodo, and die under the sun.
So what's the idea? That the Nazgûl can't see the object under the Sun, just its shadow?
The twilight, that sort of light, happens after the Sun has gone under the horizon. Objects' shadows are at their longest then. The Nazgûl, or rather their perception, are stuck eternally in that state.
Maybe this is what happens when 'thin, sort of stretched, like butter scrapped over too much bread' finally consumes your whole being. Every object goes under the horizon and stretches into a shadow.
This also would explain why the Nazgul were easily lost under the Sunlight; for them it was disorienting, pun intended. The compasses of their perception became unable to tell them where the North was.
The Witch-King was the exception, we're told. Apparently he could perceive the world of light in some way even from the twilight under which he walked.
Which brings Glorfindel to mind. He inhabiting the seen and unseen. Did the Witch-King, or rather the man who became the Witch-King, have a trace, a touch, of elvish blood? In that case he would have been not only a numenorean, but a noble or prince related ultimately to Elros himself.