In this Foster Seeley discriminator, what is the purpose of capacitor C3?
C3 provides the reference signal from the primary of the transformer to be combined with the signal from the secondary of the transformer that resonates at the centre frequency of operation. The two diodes then rectify the combination.

At the centre frequency the primary and secondary signals are in phase and the net result is zero output. As the frequency deviates from the centre the phase of the secondary changes relative to the primary. As a result the voltage at the output of the diodes changes positive or negative depending upon the direction of the frequency change.

C1, C2, R1 and R2 are selected so that D1 and D2 rectify the RF signal to provide a voltage proportional to the amplitude of the summation of the signal through C3 and the voltage across the secondary. The time constant of C1R1 and C2R2 should be much longer than the RF period but short relative to the highest modulation frequency.

It looks like this diagram was taken from this website that describes the operation:
Foster Seeley Discriminator

A more detailed description is here FM demodulation