D1 is the voltage reference zener. Tied to this is a string of divider resistors (R2-6) that set the various fixed voltage levels. R7 & 8 form a voltage divider to that reduces the battery voltage by a factor of 3. U1 is an LM339 quad comparator that compares the various voltages from the two dividers. The comparator sections have open collector outputs that simply function as switches to operate the LEDs. D7 protects against reverse battery connection.
The LEDs are biased to operate at about 4mA which is quite bright if modern LEDs are used. This current can be adjusted simply by varying the series resistors (R9 through R13). The overall current drain as shown is about 25mA which tends to be wasteful for continuous operation. For energy conservation, connect to battery via a pushbutton (Push to Test).