To put it very generally, the "honest services" theory allowed a much broader theory of criminal activity than those used in cases of bribery itself. Basically, if I'm working for the people of Kentucky and I'm in effect in the pay of a private interest, I'm depriving the people of the state of "honest services" even if the prosecutor cannot prove, narrowly speaking, that I took a bribe. In short, it makes it much easier for prosecutors to make their case.
The Court unanimously decided to scale back "honest services" and the conservative wing (Scalia, Thomas and Kennedy but not Alito) wanted to get rid of it altogether.
If you've followed TPMMuckraker over the last five years, I'd venture to say that the majority, probably the great majority of the public corruption cases we've covered relied in whole or in part on "honest services." So if it's been dramatically curtailed that could undermine a lot of convictions.
There's been surprisingly little commentary published so far on just how far-reaching the decision is, probably because it's a very complex decision and the experts are taking some time to digest it. We'll bring you more over the course of the day.