Working of a Diode Ring
In this circuit, every diode acts similarly to a switch. Diodes are used because they switch fast (ten million switches per second). In the simple model, the switch is either open or closed.
The local oscillator determines switch timing: on one half cycle both diodes are open, on the other half-cycle both diodes are closed.

The circuits below are a slight modification of the OP's diagram. Works similarly, but may be easier to see the functional operation:







simulate this circuit &ndash; Schematic created using CircuitLab
 When L2 and L3 are equal, and D1 matches D2, the local oscillator signal (which is switching diodes D1 &amp; D2 on and off) doesn't disturb the signal....it is a balanced bridge. If balanced, no local oscillator signal flows through Rload. During the half-cycle when the two diode switches are closed (on), Vsig current can flow through Rload. Half of this current could flow through L2,D1, and half through L3,D2. During the half-cycle when the two diode switches are open (off), no Vsig current can flow.

For one-half cycle, Rload is connected to Vsignal, and for the other half cycle, Rload is disconnected from Vsignal. Notice that no ground is shown. Ground could be placed at the junction of L2,L3,Rload, or ground could be placed at the junction of D1,D2,Vsig...the only difference is in the phase of the output signal across(through) Rload. In the OP's diagram, ground is placed at the junction of L2,L3,Rload.

If the frequencies of Vlocal osc and Vsignal are the same, then the average voltage (or current) in Rload has a DC component which depends on the phase relationship between Vlocal oscl and Vsignal. This circuit can be used as a phase detector.
If the frequencies of Vlocal osc and Vsignal differ, then the voltage (or current) in Rload has a number of component parts, some of the major ones being:


Vsignal frequency
Vsignal + Vlocal osc frequency
Vsignal - Vlocal osc frequency


This waveform example shows the output voltage or current at Rload where Vlocal osc frequency differs from Vsig frequency. One-and-a-half cycles of the low-frequency difference (Vsignal - Vlocal osc) can be clearly seen.
This simplified mixer is only half-balanced. This means that very little of the switching signal of Vlocal osc appears in Rload. There are variations of switched diodes that short Vsignal for one-half cycle, instead of this variation where Vsignal is opened (off) for one-half cycle.
 A full-bridge of 4-diodes conducts signal current on both half-cycles, but reverses Rload current direction on alternate half-cycles of Vlocal osc. This version can be fully-balanced, where Rload is isolated from both Vlocal osc and from Vsig, and sees mostly the two mixing frequency components Vsig + Vlocal osc and Vsig - Vlocal osc: