Looks like the idea is similar to that of a computer keyboard. In a keyboard, the controller would apply a high voltage to a row, and read back the voltage at each column. If you only set one row high at a time, you know which switch is causing its column to go high. Comparing this to your device, a "row" would be [A, B, or C] and a "column" would be [a, b, c, or d].
The example given in your second image seems to indicate this. Set the switch to position A. Then set B high. You'll observe both a and b are high (because S3 and S4 are closed), c is high or low (because S10 is either open or closed), and d is low (because S9 is closed). Since you're not setting C high yet, S7 and S8 play no role.
You can extend that example and consider what happens when you set B low again, then set C high. What outputs would you observe as high, assuming your switch is still in position A? (Answer: d only)
Your controller should set each of A, B, and C high independently, observe the outputs, and compare to the chart. A, B, and C will be outputs of your controller, and a, b, c, and d should be inputs.
I would recommend a series diode at each of A, B, and C. Without diodes, consider what happens if you set B low and C high while switches S4 and S5 are both closed. You will have your "high" shorted directly to "low".
Weak pulldown resistors on the outputs would be prudent, to ensure a good logic-low when the connected switches are closed.