If I have correctly understood, its basic working principle is the fact that at the secondary coils (L2) of the Balun there will be two signals equal and opposite. They are added to the signal applied to L1 through C8. This means that the 2 diodes will receive:
+V2/2 + V1 and -V2/2 + V1
During the instants in which the RF input's frequency is equal to the carrier, C1 and C2 are sized to realize resonance and so V2 phasor will be orthogonal to V2 phasor. This, in addition to the previous relationship, will mean that the signals applied to the diodes will have the same absolute values. And so? Which is the role of the diodes?
Reference: https://www-elec.inaoep.mx/~rogerio/DemodulationGeneral.pdf, page 13