The circuit you linked to will work. It is attached.
Anytime we want to measure temperature for any number of reasons we start with a device to sense the temperature. Boncuk's circuit uses a LM35 temperature sensor which is a very good choice. The circuit you linked to uses a 'thermistor" as a temperature sensor and it will also work. However, the circuit has one drawback as drawn that I can see. Note Boncuk's circuit includes what we call Hysteresis and the latter circuit does not.
Hysteresis is important in circuits like this that turn a fan on and off based on temperature.
Give this link a read. Pay attention to the section about Hysteresis.
As temperature rises we want the fan to turn on at a given point for cooling. When cooling begins the temperature will drop and the fan will turn off. This can result in the fan constantly turning on and off over and over again. So based on your 25 C temperature if the temperature exceeds 25 the fan is on and below the fan is off. This would work better if the fan came on above 25 and turned off at maybe 22 or 23 C. For your circuit a 100KΩ resistor placed between Pin 7 output and Pin 2 + Input of the comparator would give you some Hysteresis.
Read the link and try to gain understanding of how a comparator circuit works. Then you should understand how your circuit actually works.
Ron