How it's meant to work:

1. In the entire circuit, there are four 'rows'.
2. Each 'row' has three 'cells'.
3. Each 'cell' has an LED and a n/o pushbutton switch.
4. The controlling lines:
   • The top line is capable of sourcing enough current to power a whole 'row' of LEDs, and can be switched high or low.
   • The bottom lines are general-purpose I/O ports on a microcontroller, and can be set as high/low outputs, or high-impedance (input).
5. The top line is high when displaying LEDs, and low when reading buttons.
6. The microcontroller can set which row it is 'looking' at by setting which ONE of its lower 4 lines high.
7. To read all the buttons, the microcontroller sets the top line low, then:
   1. Sets the first of the bottom lines HIGH, and the rest to inputs.
   2. Reads the button state (1=on,0=off)
   3. Sets the second of the bottom lines HIGH, and the rest to inputs.
   4. Reads the button state (1=on,0=off)
   5. ...
8. To display a LED pattern, the microcontroller sets the top line high, then rapidly:
   1. Sets the first of the bottom lines HIGH.
   2. Sets the remaining lines to LOW if lit, high-impedance if not lit.
   3. Pause for a brief moment.
   4. Sets the second of the bottom lines HIGH.
   5. Sets the remaining lines to LOW if lit, high-impedance if not lit.
   6. Pause for a brief moment.
   7. ...
   8. Repeat as long as the pattern should be shown

If you're quite confused, read this about charlieplexing - the technique this is derived from.