Here is where the magic happens: the I and Q synchronous detectors!
Here the chroma signal arrives at the detector's grid, where it is combined with the signal from the local subcarrier oscillator elsewhere in the schematic. The 2 detectors combine the subcarrier and chroma to provide an output to the phase splitters, which provide positive and negative I and Q signals for use in the matrix which drives the CRT. What's actually happening here is quite technical, but to sum it up the output of the pair of detectors will vary depending on the degree of phase shift between the incoming chroma signal and the local oscillator. This output voltage is a direct representation of the hue of the incoming signal.