Can someone explain how this Ratio detector circuit works?
The diagram in your question comes from a NEETS (US Navy Electricity &amp; Electronics Training Series) module, to be more specific it comes from module nº12 . The documentation for these modules seems to be freely available in the internet from several sources. Here as a PDF, or here as a webpage.

You can find a detailed discussion of circuit operation in pages 3-15 to 3-17. The discussion is focused on how the circuit operates above, below and at resonance. You should read the full discussion, but I'll give some short answers here to your questions:


The RATIO DETECTOR uses a double-tuned transformer to convert the instantaneous frequency variations of the fm input signal to instantaneous amplitude variations. These amplitude variations are then rectified to provide a dc output voltage which varies in amplitude and polarity with the input signal frequency. This detector demodulates fm signals and suppresses amplitude noise without the need of limiter stages.
Diodes CR1 and CR2 work as rectifiers. They block the negative part of the FM signal cycle. 
The third transformer winding L3 provides additional inductive coupling which reduces the loading effect of the secondary on the primary circuit. You get an in-phase sample of the input voltage without having to load the primary.
C5 is a smoothing capacitor. It keeps the voltage approx constant when CR1 and CR2 are reverse biased during the negative part of the FM signal cycle. This can be achieved by letting the time constant of C5, R1 and R2 be much greater that the FM signal period.
Resistor R3 limits the peak diode current and furnishes a dc return path for the rectified signal.
R5, C6, and C7 form a low-pass filter to the output.
C8 is a coupling capacitor. It AC couples the demodulated signal to the next stage.