One reason could be a sense of disillusionment with the village's leadership. Maybe they discover some dark secrets about the elders and decide they can't be part of it anymore. Another reason might be to protect the village in their own way. They could think that by leaving, they draw danger away from their home. Also, they might be seeking a greater power or knowledge that can only be found outside the village.
In some fanfictions, Naruto and Sasuke leave the village because of a prophecy. They believe that they are meant to go on a journey outside to fulfill a greater destiny. It could also be due to personal growth. They feel restricted within the village and want to experience the wider world to become stronger. Sometimes, it's about revenge too. If someone from outside the village wrongs them and the village doesn't take proper action, they might leave to seek justice on their own.
There are fanfictions where Naruto and Sasuke leave the village because of a rift between them and their peers. They might be misunderstood or misjudged by the other villagers. For example, if Naruto is wrongly accused of a crime he didn't commit and Sasuke stands by him, they could decide to leave rather than face the unjust treatment. Or perhaps they are tired of the constant fighting and politics within the village and simply want a fresh start away from it all.
Chapter 315, "The Strange Cold Apple Tree and the Seized Silver Mine", is the clearest early answer because it turns Cold Apple Tree's situation into a concrete plot event. The chapter gives readers a real answer by opening on I heard from Hou Hu that out of the two Cold Apple Trees, one was... and ending on Unfortunately, Xia Hong completely ignored her, waiting for Li....
Chapter 308, "Shocking Turn, Grave Injuries, and Restless Hearts", is the clearest early answer because it turns Yanhan Blade's situation into a concrete plot event. The chapter turns that into something concrete by moving from Five people emerged from the crowd. to But judging by the loud noise when they hit the ground, those who....
Chapter 58, "Jiang Yan Stranded Abroad", is the clearest early answer because it turns Despite Director Sun's situation into a concrete plot event.
Chapter 283, "Breakthrough, New Army, High-Intensity Training (Part 2)", is the clearest early answer because Beast Meat finally turns buildup into a real advancement. Early on, In Yang Zhong's eyes, there was a longing for Great Xia.. By the end, Seeing Yang Zhong jump down from the tree, Li Hu could obviously feel....
Chapter 323, "Approaching Labor, the Talented Xiao Family Assembles Urgently", is the clearest early answer because it turns Xiao Kangcheng's situation into a concrete plot event. Early on, Second Year of Great Xia, December 15 Not long after nightfall, Inner.... By the end, Xiao Kangcheng first gave his son, Xiao Hen, a glance, blaming him.... Inside the broader power growth storyline, the chapter closes with Hearing Xiao Ning's words, Lin Lan retorted, "How can that be the same?
Chapter 316, "Blood Jade Fox and Teasing, Boiling Xia City, Migration Begins (Part 3)", is the clearest early answer because it pushes Blood Jade Fox into a new phase or setting. The chapter turns that into something concrete by moving from Li Xuanling and Li Tiancheng both walked over from behind at this... to Li Xuanling asking about the price of Yangyuan Pill proved she had....
Chapter 119, "Wen Jiaojiao is Poisoned!", is the clearest early answer because it turns Director Sun's situation into a concrete plot event. The chapter turns that into something concrete by moving from What is Princess Wen talking about?
Chapter 306, "Helplessness, Shifting Loyalties, Total Annihilation, and New Territory", is the clearest early answer because it turns Li Xuandu's situation into a concrete plot event. Early on, Year 48 of Longyou, June 26 Four more hours until daylight.. By the end, They weren't foolish, once they were out of the ground, they quickly....
Chapter 331, "Zhu Shun and Huang Tianxi, the 3rd Year Cold Yuan Banquet of Great Xia, and Xia Yuzong (Part 5)", is the clearest early answer because it turns Li Tiancheng's situation into a concrete plot event. Early on, The children immediately lowered their heads in fear, Huang Tianxi.... By the end, The Seven Departments in the Outer City East, South, West, and North....
Chapter 109, "Not One Deserves to Stay on His Friends List!", is the clearest early answer because it turns Uncle Zhao's situation into a concrete plot event. The chapter turns that into something concrete by moving from And another thing," Jiang Li said, taking out a tissue to wipe her...