I have assembled the attached adjustable dual power supply kit but need a thorough explanation of how it works/operates (i.e. how do the components affect the behavior of the device). Thanks in advance

 

The regulators are LM317T and LM337T and you will get most details from their datasheets and example circuits.

 

The transformer's 12.6Vac output is converted to approximately plus and minus 17Vdc by diodes CR1 and CR2 respectively, and filtered by C1 and C2.
These are regulated by adjustable regulators IC1 and IC2 to give the desired DC output voltages, and filtered by C3 and C4.
CR3 and CR4 protect against accidental reverse voltage being applied to the inputs.

 

The transformer's 12.6Vac output is converted to approximately plus and minus 17Vdc by diodes CR1 and CR2 respectively, and filtered by C1 and C2.
These are regulated by adjustable regulators IC1 and IC2 to give the desired DC output voltages, and filtered by C3 and C4.
CR3 and CR4 protect against accidental reverse voltage being applied to the inputs.

How can it have 12.6Vac output?

 

So this is basically a half wave rectifier?

Yes, it's a cheap design with two half-wave rectifiers which generate higher ripple voltage and undesirable DC current in the transformer (which can cause core saturation), as compared to a full-wave rectifier.
It would be much better to use a 25.2Vac center-tapped transformer with a bridge rectifier across the 25.2V and the center-tap grounded, to effectively form two full-wave rectifiers generating the two DC voltages.

 

How can it have 12.6Vac output?

I don't understand the question.

 

I don't understand the question.

Whoops I read that wrong. My apologies...finals are frying my brain.