The most useful lesson about writing I learned I did actually learn in school (in a mandatory writing-intensive course in college): to re-read and edit.

I think it was specifically that reading what you wrote aloud (or mouthing it silently) is a way to get a fresh look at what you wrote. It helps you realize both 1) more natural sounding and 2) more logical ordering of words than the stream of thought you wrote down.

My initial stream of thought when writing is always a garbled mess. So this technique made a big difference for me.

The most that my favorite books on writing (mentioned in the article as well: Zinsser's On Writing Well) helped with was just cutting out unnecessary crap.

Not to say it doesn't help to read these sorts of books. And I agree with the author that just writing (and continuing to write as practice) is best.