The way I understand it, there's hardware in every bluepill/redpill's brain that connects the human to The Matrix. It's interpret The Matrix code to biological impulses in our human body. Pain, pleasure, cold, hot, etc.
When Neo went to The Architect, he touched The Source, the source code for not only The Matrix but also for Zero One. When he touched The Source, he became a part of the network that machines use for not only bluepills, but for every machine in Zero One, and it works because of that hardware.