One possible reason could be his pursuit of power. Naruto might feel that he can't achieve his full potential in the current place and needs to go on a journey to become stronger. Another reason could be to fulfill a promise or a sense of destiny. He may have a calling that he believes can only be answered by leaving.
Naruto could be driven by a thirst for knowledge. He might have heard about certain jutsus or powers that are only available in other places. He also may feel restricted by the village's rules and regulations and leaving gives him the freedom to explore and grow in ways that are not possible if he stays.
Perhaps in this fanfiction, Yugito was involved in a political intrigue within the Hidden Cloud Village. Naruto, being an outsider with influence, banished her to maintain peace between different factions. Yugito's actions or associations might have been such that it was the only way Naruto saw to prevent a civil war or major conflict in the village.
Sasuke could end up in a very dark place. He might get involved with some really bad characters or groups that could lead him further astray from his original path. He could lose his sense of self and become consumed by revenge or whatever negative emotions are driving him to leave in the first place.
One reason could be to seek greater power. Maybe they feel that Konoha no longer offers the training or resources they need to reach their full potential. So they decide to leave and explore other places where they can become stronger.
It could also be due to Naruto's outcast status in the village initially. Danzo might think that banishing Naruto would be an easy decision as Naruto doesn't have many supporters at first. He might believe that by getting rid of Naruto, he can maintain the status quo in the village and prevent any disruptions that Naruto's presence might cause in the future.
It could lead to a major shift in the relationship dynamics. For example, it might create a rift between Sakura and Naruto that will be hard to mend. This can add a lot of emotional depth and conflict to the story.
One possible reason could be to search for a new power source. Maybe in the fanfic, the powers within the Elemental Nations have become stagnant for him, and he believes there are greater sources of power elsewhere.
One possible reason could be his strong bond with Naruto. Their relationship has been through so much, and Sasuke might realize that Naruto understands him on a deeper level than Sakura. Maybe he feels that his journey and growth are more intertwined with Naruto's, and he values that connection more in this fanfiction.
One possible reason could be Naruto's over - eagerness to prove himself. Naruto is always trying to show that he's strong and capable, and sometimes this might come across as him not considering Sakura's feelings or opinions. For example, he might rush into a dangerous situation without consulting her, which could lead to an argument as Sakura is also a capable ninja who wants to be part of the decision - making process.
If Sakura is overly critical of Naruto in the fanfiction, it could push him away. For instance, if she constantly belittles his dreams or his progress as a ninja, Naruto might feel that he's not wanted or respected around her and thus decide to leave.
One reason could be that the writer has outgrown the Naruto fandom. As people grow and change, their interests shift. They might find other anime or stories more appealing now.