The dissent (Scalia, Alido, and Thomas) believe that Aereo did not "perform" at all. They believe that when an Aereo subscriber logs in and clicks the play button, that the subscriber, not Aereo, is doing that performing.

EDIT: Please note that the dissent would not have necessarily ruled that Aereo's service was legal. The dissent merely believes that Aereo's service complies with this one specific part of the Copyright Act. From the dissent: "[Our] conclusion does not necessarily mean that Aereo's service complies with the Copyright Act. Quite the contrary. The Networks' complaint alleges that Aereo is directly and secondarily liable for infringing their public performance rights and also their reproduction rights."

So this decision is designed to be as tight as possible. The Court is trying to limit the effects of its decision by constraining it to this one claim by the networks. From the language of the dissent, I have a hunch that even if this one aspect of the case had been decided in Aereo's favor, that some other aspect would have resulted in its service being declared illegal.

Scalia: "I share the Court's evident feeling that what Aereo is doing (or enabling to be done) to the Networks' copyrighted programming ought not to be allowed."