The whole decision is much worth reading, and, except for the standing issues, quite nontechnical. It's also hard to boil down, since the argument's components are closely integrated. Still, here is what seems to me to be the best very short excerpt:

In determining whether the Second Amendment’s guarantee is an individual one, or some sort of collective right, the most important word is the one the drafters chose to describe the holders of the right — “the people.” That term is found in the First, Second, Fourth, Ninth, and Tenth Amendments. It has never been doubted that these provisions were designed to protect the interests of individuals against government intrusion, interference, or usurpation. We also note that the Tenth Amendment — “The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people” — indicates that the authors of the Bill of Rights were perfectly capable of distinguishing between “the people,” on the one hand, and “the states,” on the other. The natural reading of “the right of the people” in the Second Amendment would accord with usage elsewhere in the Bill of Rights.

The District’s argument, on the other hand, asks us to read “the people” to mean some subset of individuals such as “the organized militia” or “the people who are engaged in militia service,” or perhaps not any individuals at all — e.g., “the states.” These strained interpretations of “the people” simply cannot be squared with the uniform construction of our other Bill of Rights provisions....

The District points to the singular nature of the Second Amendment’s preamble as an indication that the operative clause must be restricted or conditioned in some way by the prefatory language. However, the structure of the Second Amendment turns out to be not so unusual when we examine state constitutional provisions guaranteeing rights or restricting governmental power. It was quite common for prefatory language to state a principle of good government that was narrower than the operative language used to achieve it.

We think the Second Amendment was similarly structured. The prefatory language announcing the desirability of a well-regulated militia — even bearing in mind the breadth of the concept of a militia [which the court had earlier concluded “was a large segment of the population” rather than just a government-selected National Guard-like subgroup -EV] — is narrower than the guarantee of an individual right to keep and bear arms. The Amendment does not protect “the right of militiamen to keep and bear arms,” but rather “the right of the people.” The operative clause, properly read, protects the ownership and use of weaponry beyond that needed to preserve the state militias....

[I]f the competent drafters of the Second Amendment had meant the right to be limited to the protection of state militias, it is hard to imagine that they would have chosen the language they did. We therefore take it as an expression of the drafters’ view that the people possessed a natural right to keep and bear arms, and that the preservation of the militia was the right’s most salient political benefit — and thus the most appropriate to express in a political document.