Why do they make traces to LTZ1000 in spiral shape?
It is to reduce the thermal gradient across the device.

A longer meandered track will carry less heat to and through the part than a short straight track. Note also that the PCB substrate has been milled away between the tracks; the PCB probably conducts most of the heat.

We normally think of a PCB as performing mainly the electrical function of connecting the parts together, and the mechanical function of holding them securely. As the manufacturing process is simple, reliable and accurate, PCBs are also useful for simple mechanical engineering tasks like this.

The datasheet says:


  Thermocouple effects are one of the worst problems and can give apparent drifts of many ppm/°C as well as cause low frequency noise. The kovar input leads of the TO-5 
  package form thermocouples when connected to copper PC boards. These thermocouples generate outputs of 35µV/°C. It is mandatory to keep the zener and transistor 
  leads at the same temperature, otherwise 1ppm to 5ppm shifts in the output voltage can easily be expected from these thermocouples.


So the elaborate board design seems specifically to counter this thermocouple effect. The thin leads and cutouts increase the thermal resistance from rest-of-board to the device, and the circular patterns near and under it try to keep the footprint a highly conductive region.