Could someone help me learn here, by walking me through the operation of the circuit?
Before the circuit above is powered up, all the capacitors will have zero volts across them. When 24Vac is applied to the input, it is rectified and it's peak voltage of 33V charges up the 470uF cap in a very short time (milliseconds). The 100uF cap charges very slowly through the 180K resistor and it is this charge time that sets the delay. Because the 100uF has zero volts across it upon power up, pin 2 of the OpAmp will have nearly 33V on it. Since this is much higher than any possible voltage on pin 3 (11-22V) pin 6 of the OpAmp goes to near zero. This forces the voltage at pin 3 to stay at 11V. As the 100uF cap charges, the voltage at pin 2 of the OpAmp will drop. Once it drops below 11, pin3 will now be greater than pin2 and pin 6 of the OpAmp will go up to nearly 33V. This will cause the voltage on pin 3 to change to 22V (due to the 100K feedback resistors) ensuring that the OpAmp switches with a snappy hysteresis action and stays on. It also supplies base current to the 2N2222 which turns it on along with the relay as well.