How does this LED setup work?
Functional Reasons

Placing LED's in a parallel-series combination like this is a technique to average out the brightness given constant voltage. It helps present a more even looking illumination cheaply (as opposed to active constant-current circuitry). 

The Q-point of the two LED's in parallel will be determined by the voltage assigned to the pair by the current allowed through the 4 series steps.

Diodes in Parallel


All diodes have a forward voltage drop that is non-linearly related to current. 
This curve is not identical even for diodes made in the same batch


The final Q-point will be V1 = V2 (diode voltages equal), I1 ≠ I2 (currents unequal).

However, in your case the parallel combination is itself in series.

That means that I1 + I2 = Ip is the same for each parallel pair of diodes in the same string.

Light Emitting Diodes

In the case of LED's in the forward region, brightness is almost linearly related to current. So holding Ip constant gives you approximately the same amount of light from the pair of diodes even though it is difficult to know how the light output will be distributed between them.

Extending this to the string of pairs, you cannot find a closed form solution (I might be wrong, but I don't see one) so you would need to solve it iteratively (I like Newton-Raphson) like SPICE does.

In general, however, the more diodes in each parallel group and the more parallel groups in each string, the closer the light output of the parallel groups approach each other given a batch of LEDs, normally distributed, by light output.