OK, I read it. For background, the original statue reads:

It shall be unlawful foró

(1) a governmental entity to sponsor, operate, advertise, promote, license, or authorize by law or compact, or

(2) a person to sponsor, operate, advertise, or promote, pursuant to the law or compact of a governmental entity,

a lottery, sweepstakes, or other betting, gambling, or wagering scheme based, directly or indirectly (through the use of geographical references or otherwise), on one or more competitive games in which amateur or professional athletes participate, or are intended to participate, or on one or more performances of such athletes in such games.
Essentially, paragraph 1 prevents states from legalizing sports gambling, while paragraph 2 prevents states from licensing gambling (or more precisely, it prevents licensees from operating under their licenses).

The fundamental theory (which was a 7-2 vote, with only Ginsberg and Sotomayor dissenting) is that if the Federal government wants to make something unlawful, it has to actually do so at the federal level. It can't just order the states to do so. There are supporting arguments regarding the 10th amendment, interstate commerce, and burden shifting, but the main thrust of it is that if gambling had been unlawful at the federal level, the outcome would have been different. This is not as strong a ruling as I originally though.

The interesting thing is that by a 6-3 margin (Kennedy flipped) the Court ruled that the two provisions were not severable. Assuming paragraph (1) above is Unconstitutional, should paragraph (2) survive as the best manifestation of the intent of Congress? Ginsberg says yes, for a number of good reasons, mainly that paragraph (2) is not currently in controversy (i.e. SCOTUS doesn't have to rule on it), it does reflect the intent of Congress to prevent such gambling, and that while the Constitution might limit the power of federal government over the states, the federal government absolutely has power over individual citizens sufficient to cover the subject matter of paragraph (2). In addition to supporting the legality of paragraph (1), she thinks that paragraph (2) should have stayed in force even if (1) was stricken.

I should note that Clarence Thomas agrees with her analysis of severability, but disagrees on the whole topic of severability in the first place, and so concurs instead. Thomas apparently thinks that severability is bad jurisprudence and that Unconstitutional laws should be eliminated entirely and sent back to Congress for a do-over, and indicates that this should be true for all cases and not just this one.

Pete (also notes that Senator Orrin Hatch plans to introduce legislation to make sports betting illegal at the federal level, which might just prove Thomas's point)