Ok, let’s start from the beginning on Heller:
Scalia was an originalist, meaning he viewed the constitution through the lens of how it was understood at ratification.
He also has to be consistent with precedent or have five votes to overturn precedent. Miller was, at the time, good precedent and all this whining about how “messed up” it was or how the court “had to grant it or gut the law” is pretty meaningless. The court in Miller absolutely could have gutted the law, as they did in Heller, or done any number of lesser measures. They didn’t, they ruled the way they ruled and it was good law for nearly 70 years.
The bulk of Heller is about squaring it with the prefatory clause and/or Miller.
Section 1 is simply the facts and procedural posture.
In Section 2, Scalia starts with the prefatory clause (It being necessary to the establishment of a well regulated militia...) he writes that the prefatory clause announces a purpose, not a condition precedent, and that the operative clause (the right to bear arms shall not be infringed) supplies an individual right.
He explains his view of the context of the prefatory clause, including how it came into being and the concerns that prompted. In response to those concerns, the constitution was amended to include the right to bear arms.
That right, he says, was “not intended to lay down a ‘novel principl[e]’ but rather codifi[y] a right ‘inherited from our English ancestors...” Heller at 2801. This means that the right is subject to that same originalist contextualizing, but that it’s history goes back further than the founding.
It helps to think that The Right to Bear Arms is essentially like a title of a book, rather than an ordinary phrase. The title of the book doesn’t convey the full meaning of its contents, but Congress cannot abridge the book. You get all the chapters that existed at the time of ratification. No more. No less. (For an originalist). This is how all of the Bill of Rights is generally viewed by SCOTUS’s originalist justices (which includes all of the Trump appointees.)
He then supports this interpretation with citations to documents contemporaneous to the founding, analogous state constitutional sections, scholarly interpretations, and by reviewing the court’s own precedent. This is where he draws his “authority for saying he is right.”
Distilling all of that into “prefatory clause has no teeth” is made up reductionist nonsense. To follow that by bitching about circular logic and a lack of authority is astoundingly ignorant. He supplies the authority and historical support for what he’s saying in the pages of the opinion.
Section 3 has been posted here in its entirety. It is analogous to a very brief synopsis of the book titled The Right to Bear Arms. Scalia includes citations to authority to support it as well.
Section 4 is where he applies the established rubric to the facts of the case, finding that the individual has a right to possess, (1) a firearm commonly used (2) for a traditionally lawful purpose, (3) inside your own home.