Perhaps the ANBU has a personal vendetta against someone in Konoha who is too powerful to take on while remaining in the village. Leaving Konoha gives them the freedom to train and gather allies to come back and face their enemy. They might leave secretly, under the guise of a mission, and start building their strength in the outside world, all while Konoha is left in the dark about their true intentions.
It could be Kakashi. He has a complex past and has been part of the ANBU. Given his experiences, he might have a reason to leave Konoha. Maybe he discovers something about the Sharingan or the Uchiha massacre that makes him question the village's integrity. He's a popular and well - developed character, so a story about him leaving as an ANBU would be interesting.
Another scenario is that Naruto leaves because he feels like an outcast with the Rinnegan. The villagers might fear his new power, and he decides to leave to find a place where he can fully understand and control the Rinnegan without causing panic. He could then encounter other powerful ninjas or groups who either want to use him for his power or help him master it.
Madara Uchiha could also be the main character in this type of fanfiction. Although in the original story his actions are mostly related to his grand plans for the Ninja world, in a fanfiction, it could be explored what would happen if he decided to leave Konoha specifically because of his Rinnegan and the power it holds.
One possible reason could be a search for power. The character might feel that Konoha doesn't offer enough resources or training opportunities for them to fully explore the potential of the Rinnegan. They may believe that there are other places in the Naruto world where they can gain more knowledge and power related to their special eye.
His journey could lead him to the outskirts of the ninja world, to places that are unknown and full of mystery. With the Rinnegan, he might be able to access hidden paths or dimensions. He could be exploring these uncharted areas to gain more power and understanding, away from the politics and rules of Konoha.
One implication is that he becomes much more powerful. The Rinnegan gives him access to powerful abilities like the Six Paths techniques, which would change the power dynamics in battles in the fanfiction. He could now face stronger enemies more easily.
In Naruto fanfiction, the Rinnegan in Naruto is often portrayed in really cool and powerful ways. Some fanfics might give Naruto the ability to use all the Rinnegan powers to their fullest extent right away, like being able to effortlessly summon the Gedo Statue or use the Deva Path powers without any drawbacks. It's a way for writers to explore new storylines and make Naruto even more of a badass than he already was in the original series.
Another possibility is that it's a result of some strange chakra experiment gone wrong (or right). Scientists in the Naruto universe are always dabbling in chakra research. Maybe an experiment meant to enhance her natural abilities instead gives her the Rinnegan, and she then has to learn to control it quickly to avoid being consumed by its power.
If Naruto got the Rinnegan, he might become more cautious. The power of the Rinnegan is so great that he would realize he needs to be more careful with it. He wouldn't be as brash as before.