The four wires running to the connector are ground/power and left/right audio. I can tell for sure that the red wire is left or right, because it only runs to the second board. Orange is the other channel since it stops at the first board. That would leave yellow and brown as power, take a closer look at the marking on the board and see if they say which one is positive or negative. I see a little "C" there by the brown wire going off to the second board, that may be ground (common). According to the datasheet the chip is designed to run between 5-16V, so it may be tied a voltage regulator circuit elsewhere within this range.
I'll have to get in there and look at the circuit and measure things to know for sure what's going on (I don't have a schematic) but as with any LED meter using an IC the audio signal is connected to a series of comparators that all have one of their inputs strung together with higher and higher value resistors. The other input on them are all tied to some reference, like ground or a steady voltage. So as the audio level climbs, more and more comparators will activate, since the difference between the reference input and the audio signal input is increasing. As the signal climbs it overcomes higher and higher value resistors tied to the audio input of the comparators. If you think it might help better understand, I can post some images showing this in more detail.
All of these driver ICs have a slightly different method of configuring, this one allows you to use an external resistor to set the LED current. It also has a built-in input amplifier so you can easily adjust the meter sensitivity. Conion seems to have calibrated it so that the meters represent line-level at what, maybe around 15 or so for full-strength FM stations? The meter also has some form of peak detection/decay so those may be what one or more of those capacitors are used for.
If you want to wire it so that they bounce the same regardless of volume you'll have to rewire them to the line out signal instead, but that may involve changing one or more parts on the meter board so that they work correctly with the different signal level and impedance.
What I don't know is if the signal the C100 uses to feed the meters has been adjusted for high impedance inputs or if it's tapping directly off the speaker outputs, in which case the meter circuit may need a higher value input resistor to work with line out if that's something you want to do.