There's something I don't understand about this. My impression is that grandmasters don't play quickly (against each other). I'd be surprised if it was uncommon to spend more than five minutes on a move, for example.

So what takes so long? I assume their process is roughly "consider all possible moves, decide how good they are". So if they only look one move ahead, that sounds to me like when they evaluate a potential position, it can take more than a few seconds and they do it without reading possible future moves. Which seems unlikely.

Am I misinterpreting "one move ahead"?

I wrote a lesson plan that was used in a chess camp this summer, and one section covered various thought processes. Here is an excerpt from that section.

"As time goes on, your thought process will start to be based more and more on your positional intuition. Instead of explicitly weighing certain details against each other, you will instinctively be able to feel what direction the game is headed in. This intuition does not come naturally, but instead is the result of playing hundreds and thousands of chess games. Over the course of all those games, you will see positions similar to the one you are playing right now, and even though you may not explicitly internalize them, your subconsciousness will recognize the themes from before, and you will have some vague recollection or understanding of the position based on your prior experience."

The best answer to that that I know of is by http://en.wikipedia.org/wiki/Adriaan_de_Groot: "De Groot found that much of what is important in choosing a move occurs during the first few seconds of exposure to a new position. Four stages in the task of choosing the next move were noted. The first stage was the 'orientation phase', in which the subject assessed the situation and determined a very general idea of what to do next. The second stage, the 'exploration phase' was manifested by looking at some branches of the game tree. The third stage, or 'investigation phase' resulted in the subject choosing a probable best move. Finally, in the fourth stage, the 'proof phase', saw the subject confirming with him/herself that the results of the investigation were valid."

IIRC, in phase one, experts see wo or three moves to look deeper into. Non-experts see many more.

I recommend reading "het denken van den schaker" if this interests you. It is very readable by laypersons (and a translation in English exists)

I doubt this quote is to be taken literally. I rather think he means that good chess players are very, very good at deciding which moves to think deeper about based on intuition, hence reducing the need to visualize.

He's being witty, is all.

I think it was more of a poetic way of expressing it. That they consider many moves and their impacts, but it's based on experience, intuition, intelligence and knowledge, pruning out that bad branches quickly on, and identifying and focusing on promising ones.