This toggle circuit operates by using a couple 555 timers wired as inverters. Pins 2 and 6 are the threshold and trigger inputs to the first timer and pin 5 is the output. The output at pin 5 will always be the inverse of the input at pins 2 and 6. Likewise, the output at pin 9 of the second timer will always be the inverse of the input at pins 8 and 12. A 100K resistor connects the output of one inverter to the input of the other so the state of one will be the opposite of the other. In operation, the 1uF capacitor will charge to whatever voltage is present at the output on pin 5. When the button is pressed, the capacitor voltage will be applied to the input of the other timer which will reverse the state of both timers and toggle the relay, either on or off.
To follow it more closely, assume the output at pin 5 is +12 volts and the second output at pin 9 is zero volts. The 1uF cap will be charged to 12 volts. When the button is pressed, the cap will apply +12 to the inputs at pin 2 and 6 which will cause the output at pin 9 to go to zero, turning off the relay. When the button is released, the cap will discharge to zero, since the voltage at pin 5 is now zero. When the button is again pressed, the capacitor will apply zero to pins 2 and 6 causing the output at pin 9 to switch positive and engage the relay, and the cycle repeats.