The Network is a cloud-based server, in which B-12's (Scientist) Memories and consciousness reside.
When B-12 was a Human, he tried to upload his consciousness into a computer/robot body using a Sarcophagus because he was sick. Something went wrong and he ended up in the Network for several hundred years.
From the Network, he could observe the communication of Companions, and remembers that they created their own language after the extinction of mankind. For some reason, he could not get in touch and ask them for help with getting the body while they were living in a still safe Dead City, but when the Cat fell into the city, he was able to contact him using Cameras and monitors installed in the Dead City and its abandoned apartments.
The game doesn't say much about the technical component of the Network. B-12 describes staying there as "It was dark. I was alone." He feels the same when he is disconnected for a while due to breakdowns. His memory was also damaged while he was inside, so during the game he needs to restore his human Memories.
One theory says that after his shutdown at the end of the game, B-12 did not die, but was loaded into the Network again, since the game ends with a flash of one of the screens connected to the Control Room.