It's in the second-to-last paragraph,

Likewise, when the justices finish diagramming the Second Amendment, they should end up with something that expresses a causal link, like: "Because a well regulated militia is necessary to the security of a free state, the right of the people to keep and bear Arms shall not be infringed." In other words, the amendment is really about protecting militias, notwithstanding the originalist arguments to the contrary.

Why are the "other words" a sound restatement of the preceding words? The amendment, after all, continues to speak of the right of the people. The goal, in the author's "Because" statement, may be protecting the "militia" (not the "militias," actually, but more on that, I hope, tomorrow). But the means that the Framers used to serve the goal — even in the author's "Because" version — is protecting the right of the people to keep and bear arms.

Nor is that particularly surprising, given that "militia" referred to the able-bodied adult white male citizenry, which is to say basically most of what the Framers saw as the full-fledged citizenry, and not far from "the people." (It would have been nice if the author acknowledged that, given that the readers of the Times might well be unaware of it.) But in any case, the amendment, even under the author's reading, is "about" protecting the armed citizenry by protecting the right of the people to keep and bear arms.