How about this for an explanation (sure hope I have it all right)
If you look a little closer on the schematic, you will see that pins 1-3 and 9-12 all direct the input signal straight from the transformer to the opamp. The added connection on pin 8 (which took me a while to find
...) adds the divider network to the circuit.
In positions 1-2 and 9-12, R2-R6 are all in series to ground. They don't load down the transformer since the sum is quite high compared to the transformer secondary impedance.
On step 4, you switch some of the attenuator into the circuit. The signal goes to pin 8, through series resistors R3-R6 but R2 is now connected to ground, resulting in -20dB voltage divider type of attenuation.
On step 5, the signal goes through series resistors R4-R6 with R2 and R3 in series to ground and the attenuation is only -15dB.
The pattern continues until you hit position 8+ where there is no more attenuation.
The interesting thing is that Tim put switch positions 1-3 as the highest gain, so the stop pin is set for the lowest overall gain at switch position 4. I can only figure this made the layout easier (any comment Tim?)
Maybe this helps you understand it some - took me a while to figure it all out.