So, your first photo of the speaker wires is solid. It has 3-cables coming in. Each cable has four conductors in it. Red, black, white, and green.
A single speaker needs two wires to make it work. Often red and black.
A pair of speakers (as you have) needs four wires to make it work. Red and black for one of them, white and green for the other one (most often).
So, those three cables with their 4 wires tied together create (as you said) 3 areas of audio. It is not 3 ZONES, because they all play the same thing at the same time. It is simply 3 areas of audio which are all on and all playing the same thing at the same time.
I recommend that if you have a old receiver or something which plays audio currently that you hook it up to one of the speaker cable sets.
I can't tell for sure, but speaker cables should use the following standard:
Red: Right positive
Black: Right Negative
White: Left positive
Green: Left negative
Look at the volume controls to see if that's how they wired it. While the standard I listed above is typical for installers, the reality is that speaker wire colors just need to match on both ends, and since you already have one end plugged into a volume control, then you may need to rewire the incoming audio to the volume control.
The second photo, of the volume control, is pretty good. I can't see exactly what order the incoming cable has its wires connected in. It should follow the standard I listed above. (red/black/white/green), but it may be different. I recommend rewiring each volume control following the standard I listed above to ensure that they are following an installer standard.
The other half of the volume control wiring is going to the speakers themselves. Red/black to one speaker red/black to the other speaker. Leave those wires in place, they look good.
You say this is 'Niles', but I can't tell how you made that determination. Niles speakers? Niles volume controls?
There is a installation manual for Niles volume controls which appears to show the exact unit that you pulled out of your wall.
Read it! It really does show the connections exactly as I'm speaking about.
http://resources.corebrands.com/products/FG01652/pdf_FG01652_manual.pdf
Now, here's the most important part:
With a two zone receiver, you can have your main 'surround area', and you can use the second zone to feed audio to the second AREA you have with 3 sets of stereo speakers. It can (and will) play the same thing in all three areas. You can turn any single area up or down as you like, but you can't listen to something different in one area from another.
It is strongly recommended, that if this is the way you want to do things, then you should get a impedance matching speaker selector. They are on eBay and about 50 bucks or so to get a 4 speaker selector unit.
https://www.ebay.com/sch/i.html?_from=R40&_trksid=p2380057.m570.l1311.R2.TR6.TRC0.A0.H0.Xniles+ss.TRS0&_nkw=niles+ss-4&_sacat=0
Niles SS-4 speaker selector is a solid product to use. It allows you to take that single input from your A/V receiver which is designed for a 8 ohm speaker load, and will ensure that the 3 pairs of speakers which you are using present themselves to the receiver at 8 ohms so no damage to the receiver occurs.
It also allows you to turn off individual rooms if you would like.
There are more advanced ways of getting audio to the rooms, such as using a multi-channel amplifier to power each room independently and then connecting a dedicated streaming source to each room (like Chromecast Audio, or Sonos). Chromecast can be cheap, Sonos is expensive.
There are also multi-source selectors which can be used from any number of companies, including Niles, which offer control through your phone. Those can cost $1,000+ to get, so budget will start to matter a fair bit.
