No. What they do is give you an antenna in the cloud. Each customer has their own antenna powered by an array. You watch the same content that someone with a rooftop antenna does.
On facebook, I did a basic outline as to why this was ruled legal:
Why Aereo is legal in an easy side-by-side comparison between traditional and digital:
1) Just as you had a TV antenna on your roof that only you can access, you have your own antenna that only you can access. Each customer has their OWN antenna. Your service fee pays for maintenance of the array your specific antenna is powered by.
2) Just as you could split your roof antenna signal as many times as you wanted throughout your house to send the signal to all your TVs, you can access your Aereo antenna with any device you own via the Aereo app. Your service fee helps maintenance of the Aereo app and keeps it updated. To clarify...keep in mind you are not so much accessing CONTENT, as you are an ANTENNA! That's the key point of this right there. You're accessing an ANTENNA that is exclusively YOURS, and sending its signal to your video device. The internet is simply playing the role of the coaxial cable.
3) Just as you could use a VCR to record any signal that comes off your roof antenna to watch for later, your personal Aereo cloud storage can store recorded programs to access at a later date on any of your devices that run the Aereo app. Until 2008, this could have been illegal, but Cablevision won a decision that said they could store programs in the cloud for their customers' personal viewing. What Aereo does in this case is no different. Like the Cablevision situation, each customer has their OWN cloud storage. Your service fee helps the maintenance of the cloud storage.
4) The one limitation: Just as your roof antenna could only receive local stations that you had to live in the area to access, you need to prove you are in the right market for the service. This is why right now, only the NYC market is eligible for Aereo, despite its user being able to watch their programs over the internet. You have to live in the area to get its channels. This is due to FCC rules of exclusivity. Aereo is planning rollouts in numerous other cities (including my home area of Minneapolis) before the end of the year. Your service fees help Aereo to grow.