It starts with a tiny antenna about the size of a dime. This antenna can receive over the air TV broadcasts. Think of it as a tiny set of rabbit ears. Aereo sells you access to one of these antenna's. Only instead of sitting it on top of your TV with tin foil enhancements it sits at their location. You connect to the signal that antenna receives via the internet. Aereo facilities house millions of these little suckers. The key here is one antenna per user. Another key is that in order to have an account you have to live within the broadcast range of the over the air signal. So for example I can't have a New York Aereo account for New York broadcasts living in Los Angeles and vice versa.
Cable companies and networks are going absolutely ape shit over this and are claiming this constitutes a 're-broadcast' of their signal which is illegal. Hmmmm.... if it is a re-broadcast then anyone that uses a set of rabbit ears is guilty or 're-broadcasting' from the rabbit ears to their TV set. So long as the user to antenna ratio is 1:1 it is not even correct to call this a broadcast in the first place.
Broadcasting
Sending a signal from an antenna to a single system is not broadcasting (it is uni-casting, ie sending to one as opposed to many) just like it is not when using rabbit ears in your house. The difference here is the length of cable and transmission protocol used to connect your device to the antenna. In the case of rabbit ears the analog signal is sent via cable to your TV antenna input. With Aereo the signal is digitized and shipped to you via the internet which could involve a physical wire the whole way or in many cases jump over wireless links. The cable companies and networks are basically ignoring the fact the technology is based on a 1:1 basis just like a set of rabbit ears or ATSC tuner and instead focusing on the increased technological method of getting the signal from the antenna to the end user.
To muddy the waters a bit there are a couple of things that cloud the issue a bit from the simple explication above. Aereo is also selling a premium account that also incorporates a DVR (also housed at their location) which allows you to time shift over the air broadcasts without having any equipment at your house. Additionally since you can access your account from multiple devices, I suppose there is the possible consideration that if you used your one antenna to stream content to multiple devices at the same time you could run afoul of being a broadcast at that point.
Now, time shifting I think has already been through the wickets so the only wrinkle here is again the notion of Aereo hosting the DVR instead of it being at your house. Again if it is a one to one situation then the only difference is the length of the connection between your equipment and your viewing devices. As for the multiple device thing... you could conceivable split the signal from a single set of rabbit ears to multiple TVs and I do not think anyone would bat an eye. But that is much less well explored territory.
Details aside this is pretty much a new tech company running afoul of some very old and powerful entrenched interests in keeping things just they way they are. You may wonder what the big deal is... and it is money. Specifically advertising money and cable company re-broadcast agreements. You see cable companies have to pay to take that over the air signal from the networks and push it out over their cable systems. So having a company do this without having to broker a deal with the networks gets both parties noses bent out of joint. On the advertising front this method of viewing is not built into Nielson surveying methods so any Aereo customers are not going to figure into the eyeball count that determines advertising costs.
In my opinion the re-broadcast agreements do not apply. The cable companies take in a single signal from the network and broadcast that signal to all of their customers. They are not maintaining a 1:1 ratio of antenna to customers with an independent stream of content to each individual user. Apples and Oranges comparison. There would be an issue if Aereo were allowing customers to sign up out of market ie to folks physically living outside of over the air broadcast coverage. But they are not. So they have a very strong case to make that they are not providing something not already available to any user. They are just selling remote hosting of the antenna and DVR equipment. Both technologies not under question if housed at the users house. The advertising solution should be easy as well... just broker a deal for Aereo to provide sanitized viewership statistics to the rating companies and voila... no different than counting the eyeballs watching cable or typical over the air broadcast solutions. But no... as usual since this is new and may undermine someones way of making money it is causing a fuss.
Thankfully this injunction only hits a couple of markets Aereo is in. And they already have an appeal headed to the SCOTUS for a final ruling on this case. So far most federal cases have sided with Aereo. The cable companies are saying if Aereo wins then they will start setting up the same kind of solutions to break away from the need for re-broadcast deals. Networks are threatening to kill over the air broadcasting altogether in order to kill Aereo and any chance cable companies could use this method to avoid the re-broadcast deals. Fun fun fun. Don't you just love our legal system and corporate interests at work?