The basic HDHomeRun unit is a tiny little plastic box with an RF-input (for coax from antenna or cableTV), and an ethernet port. The box typically contains two TV tuners, and a microprocessor to control them.
Other devices on the same LAN/Wifi can communicate with the HDHomeRun, directing either of its tuners to tune to specific digital channels in the OTA or CableTV bands, and to stream a near-raw digital capture from those channels out over the network connection.
Applications on the other devices (eg. VLC, mplayer, MythTV, Kodi, etc.) can convert the digital broadcast stream into video on a computer/handset screen, or save the stream to mass storage for later.
So basically, the HDHomeRun is a pair of network connected TV tuners that are "driver free" for use by software on other devices.
Fancier HDHomeRun products include more/different tuners, and/or transcoding to smaller stream/file sizes within the little box.
Silicon Dust (makers of HDHomeRun) also offer a "PVR Service" which uses their tuner hardware along with small apps they can provide for handsets/PCs to implement a full Personal Video Recorder (PVR) setup without much technical know-how required.
I suspect Roger is most attracted by that last one, not that he in any way would be lacking on the technical know-how side!