Aereo, effectively violated all three provisions of the law. The charged a fee, retransmitted programming and provided the ability to record it. And never gained permission from any TV station to do so.
Let's start with the "recorded" part. That was struck down in the Sony VCR case 30+ years ago. We're all allowed to record programming.
Aereo's argument was that what they were actually doing was renting an individual antenna to a customer, which allowed the customers who live within the broadcast radius of a given channel to receive the OTA broadcast of that channel which could then be viewed on the device of their choice.
If Aereo's business model was to install an antenna on your roof, that they would rent to you on a monthly basis, which would allow you to watch OTA programming and possibly record it on another device that Aereo would rent to you, there would be absolutely no reason why Aereo would need to pay royalties to the broadcasters who transmit the over the air signals. They would be charging for the rental of equipment that allows you to receive programming that is broadcast freely over public airwaves. They would not be charging for the programming.
What Aereo did was to take advantage of new technology which allowed for extremely small antennas, which would be assigned to individual customers (which replaces the roof antenna) and used the Internet as the mechanism to carry the signal from that antenna to the viewing device (which replaces the coax from the roof antenna to the TV).
The individual antenna per customer is the "loophole" that Aereo was exploiting. When the law was written, they didn't anticipate the ability for a company to setup an antenna array that could be used in the manner in which Aereo did, nor did they consider the creating of the World Wide Web as a content delivery methodology.
As I wrote earlier, Aereo was violating the spirit of the Copyright laws. They were not violating the letter of the Copyright laws. That's what the justices who dissented were talking about in the quote that dpeters posted.