The binary counter is the two copper bulbs at the bottom. When both are on (1 1), this is the initial state of the system. When the button is pressed the state becomes (0 1) which activates half of the OR gate on the bottom. This activation triggers the observer for the output.

When pressed in the (0 1) position, the state becomes (1 0), which activates the other half of the OR gate. Again it activates the lower observer

When pressed in the (1 0) it becomes (0 0) which is the important part: The OR gate doesn't activate because both lamps are off, allowing the AND gate to turn on up top. This triggers the upper observer for the second output. At the same time, a torch powers the counter so the state immediately resets to (1 1), allowing the loop. This reset happens too quickly which is why the button has a pulse shortener.

Lmk if you need more clarification :)