Inner circuitry of 555 IC is needed to describe the operation of this circuit well.
 555 internal connection
Note:
1) voltage at non-inverting terminal of comparator 2 of the 555 IC is one-third of the supply voltage (R1-R2-R3 voltage divider circuit inside 555IC)
2) voltage at inverting terminal of comparator 1 of the 555 IC is two-third of the supply voltage (R1-R2-R3 voltage divider circuit inside 555IC)
3) Output of comparator 2 in the 555 IC has to be '1' to set the F/F and to reset it, output of comparator 1 has to be '1'
4) External voltage applied to pin 2 and 6 of the 555 IC is half of the supply voltage as two 4.7K resistors form a voltage divider circuit across the supply
This is how the circuit works: at the instance of power up, 6V is applied to pin2 and 6. Voltage level of the inverting terminal of comparator 2 is higher than non-inverting so the output is '0'. The output of comparator 1 is also '0' as the voltage of inverting terminal is higher than that of non-inverting terminal. Under this condition, pin 3 which is the output of 555 is '0' and the capacitor(0.1µF) has no charge. Transistor in the 555 is biased by the F/F so it conducts and forms a short circuit across LED and output point of this alternate on/off circuit.
First trigger: When switch SW is pressed, 0V from the capacitor is applied to pin 2 and 6 of the 555. Voltage of the inverting terminal of comparator 1 is higher and the output is '0'. In comparator 2, voltage of the non-inverting terminal is higher so the output is '1' and set the F/F so pin 3 is high and the capacitor charges up to approximately 12V. now the transistor inside the 555 is reverse biased and pin 7 is also high to give logic output '1' and the LED lights up.
Second trigger: When the switched is pressed the second time, approximately 12V is applied to pin 2 and 6 of 555. Inverting terminal of comparator 2 is higher so its output is '0'. Non-inverting terminal of comparator 1 is higher and the output gives '1' to reset the F/F. pin 3 of 555 now become '0' and the capacitor discharges completely. Process in first trigger is repeated when the switch is pressed again.
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