There are two things going on at the same time, DC offset and gain. If the left side of the 35.7K were grounded, your equation would be all there is too it. But to have an output voltage that goes negative for negative temperatures, you need to compensate for the DC output pedestal of the AD590. It sinks a current that is directly proportional to degrees kelvin, so at 0 degrees celsius the 590 is sinking 273.2 uA. The 581 provides this current through the 35.7K and R1.
The nice thing about this circuit is that you do not have to go through multiple temperature iterations to calibrate it. At one particular temperature, the gain and offset are independent. At 0C, both OP177 inputs should be at GND, and you adjust R1 to get there. If the input is perfectly balanced, the output will be 0 V no matter what the forward gain is, because 0 times anything is 0. After that, raise the sensor temperature to some other known value, and adjust R2 for the correct output.
ak