I've got the monograph by Sarah Greenough, which is one of my prized possessions - even though the binding broke the first time I opened it. There's a quote in the text from Willem De Kooning which I think sums up his work brilliantly "...break the willed articulation of the image, so that the image will grow, spontaneously and within its own structure and not my structure." I think about that quote a great deal, the last bit particularly. That free energy in his pictures - the lack of contrivance in his compositions, which still have immediacy somehow, is fascinating. Quiet and bold at the same time.
He wasn't an incredibly articulate man, which surprised me when I watched the video below, as his photographs exude intelligence. But I think he has it right in the quote you posted, that, to a certain extent, you have to just get it.
Don't know if you've seen this - http://www.youtube.com/watch?v=_LhYs5eq5nw
Half hour conversation with Callahan.