I found the manual for the
Niles ZR-4630 here. Enter the reCaptcha code and then click the "Download Manual" button.
This is not a trivial installation and I would consider calling a professional that deals in whole home audio. Looking at the volume control photo, they appear to be running stereo output to the control (4 wires going in) and then the other 2-wire pairs run to the speakers. It seems odd that so many wires were bundled together at the amp location and simply in parallel. Not necessarily incorrect as high impedance distribution amps can be wired that way, but if you have twelve rooms and a 6-channel amp, you would expect to have 2 rooms connected to each channel, not the four that are pictured, but 4 could work with the correct speaker type and volume control. If the speakers have transformers on them then they should be of the type used in multi-speaker setups.
You would need to pull one speaker down and see what make and model it is. If they used the correct speaker and volume control, then those two bundles of five cables could be correct and the fifth cable connects to the stereo output of one of the channels on the Niles. You would need to determine which of the 4 wires make up each stereo pair. The other rooms would be wired similarly but you want to have similar loads on each amp. Maybe the plan was to have 4 rooms on each of 3 channels?
According to the manual, the ZR-4630 is multi-zone / multi-source, so it should be possible to play different material in different zones. The telephone-jack shaped jacks on the rear of the unit are keypad connections. Were there any keypads installed or included? That may have been why the CAT-4 cables were installed; for connections to the keypads. If you don't have the keypads then this may be a futile exercise.