Its a bit complicated. The idea is that All-of-a-sudden Boba Fett is now chasing you. Out of the blue and when you were not expecting it. But pinball, usually, needs to be somewhat deterministic. Which means it needs to have some rules that you can follow ( somewhat : -- )
Also its a mode that needs to run when nothing else is running and it needs to prevent other things from running once it starts. This is one of the great things about this mode. Its just you and this gauntlet of shots. There are no other modes or multiballs getting in the way.
You can only start Boba Fett when nothing else is running. To start Boba Fett you need to make at least 4 solid white shots and the last two need to be while nothing else is running.
Boba Fett is a by product of the combo rule:
Most shots advance the combo rule. Each time you shoot one of these, in addition to what else you have going on, you will also start two other combo shots timing out. Today combo shots are pinkish in color. If you shoot one of these pink shots it will leave behind a solid white shot. These shots advance Boba Fett. White is Boba Fetts color.
So shooting combos creates solid white shots. Solid white shots lead to starting this mode.
TL;DR - The short answer is that it should happen for most people "out of the blue" when nothing else is running when you shoot a white arrow. Basically when you see a solid white arrow shoot it and it might start the mode.