How can I reuse this set of buttons from an old Sky cable TV box?
It would be much easier to understand if you rearrange schematic like this:



simulate this circuit &ndash; Schematic created using CircuitLab
Now you can see that by applying voltage to IO1 you can read the state of the buttons S6, S7 and S8 on wires IO4, IO6 and IO5 respectively. Same goes for other wires, except there are only 2 buttons to read instead of 3 when IO3 is active.
Note, that this is somewhat shoddy schematics, because it is possible to get wrong result if several buttons pressed simultaneously. Properly arranged diode matrix allows unambiguous reading of as many simultaneously pressed buttons as possible.
Also note, that you can jumper IO7, IO8 to D2, IO6. This way you'll be able to read all 9 buttons with same 6 wires.