Wiring for NuTone Roomside Bathroom fan
If I'm interpreting your pictures correctly, the cable you show is the one that went to the old light/fan. If that is the case unless you purchase a two function or three function switch as described in your installation instructions and run at least a 14/3 or 14/4 cable with ground you're not going to have separate controls for the light, fan and automatic humidity control.
With what looks like a 14/2 w/ground you will have to wire all three functions to your one single pole wall switch. That means when you flip the switch on, the light and fan will turn on together but you will lose the automatic humidity control feature. The black wire you see is for the light, the grey is for the fan and the brown is for the automatic humidity control feature.
The only other option from what I see is if you can run a new cable to the fan. Adding another switch doesn't help unless you have at least a 14/3 w/ground or 14/4 to include the automatic humidity control supplying the fan/light.
Edit
If you have only three wires (black, white &amp; ground) going from the wall switch to the fan/light and you can't run another cable you might want to consider returning it and getting a simple fan/light setup that will work like your old one. Right now you're paying for features like the timer and humidity control that you can't use.