LED behavior during 24 VAC negative half cycle
Your schematic shows two basic topologies: one with and one without a resistor in parallel with the switch. For each of these topologies, there are four possible states: two with the switch closed and two with the switch open. So, altogether, there are eight situations to examine.

Let's start with the cases on the left, without the resistor in parallel with the switch:







simulate this circuit &ndash; Schematic created using CircuitLab

At the far left, you can see that there is only one path, through the RED LED, to complete the circuit. So for this half-cycle, the RED LED is activated and showing. But just to the right, with the AC reversed, there is NO path available. So both LEDs are off. So for the left two cases, with the switch open, only the RED LED can be lit. And only then, for one half-cycle. Given the 50-60 Hz frequency, this is sufficient for you to perceive the RED LED as "ON" and the GREEN LED as "OFF."

Now with the switch closed, you can see on the left side of this right-pair of cases, that the current flows through \$D_3\$ rather than through \$LED_{1_\text{R}}\$. This is simply because a "normal" diode has far less of a voltage drop than does an LED. So the current does NOT flow through the RED LED, despite the fact that it is oriented with the right polarity. \$D_3\$ effectively bypasses the RED LED here. So the RED LED is still "OFF" in this half-cycle. On the far right side, with the AC reversed again, the current cannot go through \$D_3\$ or the RED LED. It can only flow through the GREEN LED. Luckily, there is also \$D_2\$ to complete this circuit. So for the right two cases, with the switch closed, only the GREEN LED can be lit. And only then, for one half-cycle. Given the 50-60 Hz frequency, this is again sufficient for you to perceive the GREEN LED as "ON" and the RED LED as "OFF."

In your last case, with the added resistor in parallel to the switch, this only affects things for the case where the switch is open:







simulate this circuit

(I've removed the inactive diodes above.)

This will appear as a YELLOW/ORANGE color to your eye, since both colors are being emitted; one in each half-cycle.

In the case where the switch is closed, the new resistor is bypassed and the behavior will be as before (a GREEN LED lit.)

So here, you will have YELLOW or GREEN. But never RED, exactly.

The exact currents in this last case are a little more tricky. But it's not terribly important. The value of \$R_2\$ can be adjusted according to need, easily.



I've not addressed the details with respect to how many of these are series-chained. But that's another detail regarding the selection of \$24\:\text{VAC}\$ and isn't quite as important. Fewer sections will mean brighter LED emissions. More sections will mean dimmer. But otherwise, the logic remains similar.

This isn't a well-managed system in the sense of the voltage drops for each section stacking up always the same. But it's probably good enough for this use. They could add more circuitry, I suppose. But they felt it wasn't necessary.