Earlier quoted context omitted.
Indeed, "write like you talk" is not quite the right way to go about it. In my experience that usually leads to a stream-of-consciousness mess. However, one thing I have found almost universally helps my students is similar: read what you have written OUT LOUD. Is it halting and hard to read? Confusing and obtuse? Then you should probably fix it. There seems to be some magic about actually speaking the words out loud…
Isn't he basically saying use your natural lexicon vs. an artificial one (typically used to make oneself sound more educated).
The advantage of reading it out loud is (I'm guessing) that you process it with a different part of your brain. When you just read it, the words are your own and you don't process it the same way, whereas when you hear it the words in a sense sound external and it is easier to hear where it sounds awkward.