Theoretically, an empty 1.5 farad(!) capacitor is zero resistance. If you apply power to such a capacitor, the zero resistance may pull a massive current from the +5V power source. The NTC thermistor prevents this massive current surge by imposing a high resistance when starting up from cold. As the NTC slowly warms up from the current passing through it, the resistance will slowly fall, gradually increasing the capacitor charging rate until it is full.
D1 prevents the capacitor from backfeeding into any circuits when main power is lost (I assume the switch is to illustrate removal of main power, rather than being a central element in circuit operation. Its placement somewhat confuses the entire understanding of the circuit operation). When main power is lost, backup power from the capacitor can flow via D2, you don't want to waste backup power in the NTC so it is bypassed.
HOWEVER
In practice the supercapacitor will not be zero resistance when empty. The supercapacitor datasheet will specify its internal resistance. And this will dictate whether the NTC and D2 is even needed in the first place, as will the impedance of the +5V main power source.
Whilst inrush current limiting is a real phenomenon (see article below), its application here seems unwarranted.
https://www.sound-au.com/articles/inrush.htmhttps://www.sound-au.com/articles/soft-start.htm