X1 reduces the 240V input down to 18V. D5 and D6 form a half wave bridge. The schematic says +/-24V, but you won't get that from a Half Wave bridge. You'll only get about half of that, so I would say, +/-12V. If you don't draw a lot of current, it may go higher, but I doubt it. D3 and D4, along with R7&8, form a +/- 15V? regulated supply for the opamp circuit. R5 and D1 form a 5.1V reference supply for the inputs. Note that the schematic says -5V. This is incorrect, it's +5V.
R1 and R3 supply a reference voltage, your temperature set point. R2 and R4 (R4 being the temperature sensor) are the input you are measuring. Thermistors are NTC (negative temperature coefficient), meaning as they warm up, they have less resistance. Colder = higher resistance.
Control input = (-) input. Measurement input = (+) input... therefore, when your temperature voltage goes above your control voltage (COLDER than the set point), the opamp output sources current, and turns on the transistor T1 (through the un-needed R6), which turns on the Relay RL1 (R1 in the parts list). When the temperature voltage goes below the control voltage, then the opamp ouput sinks current and turns the T1 transistor off, turning the relay off.
I say R6 is un-needed, as opamps are current output devices, not voltage ouput devices, therefore, the voltage floats as necessary. Transistors are current input devices, only needing base resistors when driven by a voltage output device.
I'm not as critical as some about the use of an (obsolete) opamp, as you can replace it with just about any general purpose opamp off the shelf. They use an opamp for the voltage comparator because it saves the biasing resistor to turn the transistor on (most comparators have an open collector output and pull to ground or open up). What I don't like about this circuit is that they have no hysteresis built in. Imagine your control voltage being right at your input voltage... the smallest amount of noise will cause your output to continually switch on and off, on and off.... garbage design. Also, why do they need +/- supplies? If they're worried about the temperature input voltage going too close to ground (common mode on most opamps is +7v and -5V on the inputs) then they could add a resistor to ground under the thermistor to raise it up.