If it’s not receiving radiofrequency waves, it’s not an antenna.
It's a poorly written article. I read a much better written one over a year ago.
Aereo opens an office across the street from the television station's broadcast antenna(s). Since they are so close to the broadcasting antenna(s), they can use very small antennas to pull the signal in. So it does use actual antennas to pull in radio frequency waves.
Each subscriber rents their own mini antenna in Aereo's office. A television's radio frequency signal is tuned in, digitized and streamed to the subscriber for that antenna, and no one else. It is not recorded by Aereo, though I suppose it potentially could be by the end user.
I am 15 miles from Milwaukee's OTA antenna farm on Capitol Drive. I cannot always reliably pull in every Milwaukee digital channel using the DIY Gray-Hoverman antenna I have on my roof. Subscribing to Aereo would solve that.