I don't think the author is suggesting that details don't matter. Rather, he is suggesting that on a first pass through material, it is often better to focus on learning the material on a conceptual level (where is this material taking me? What does this theorem really tell me?) rather than focusing on the mechanical details of the derivations and proofs. To a certain extent, this is already built into the curriculum: freshman and sophomore mathematics coursework tends to focus on concepts and computation, while junior and senior coursework chases after the fundamental reasons why the theory works. Consider, for example, the relationship between Calculus I (freshman course) and Intro to Real Analysis (sometimes called Advanced Calculus, typically in the senior year). These courses cover very similar material, but the mathematical maturity required for Real Analysis course is significantly higher. I believe the author is suggesting that students spend more time on understanding the "why" and less time on "how". You can go back and figure out the "how" much quicker a little later on. "How" is nevertheless important, because ultimately you need to learn how to prove your own results.
The other important point is that the author's intended audience are individuals who are determined to master mathematics at a deep level, the sort who are determined to crank through all the details. The message makes much less sense outside of the intended audience.