Perhaps he is lured away by an unknown force. In some fanfictions, there might be a mysterious character or group that tricks Naruto into leaving Konoha. They could make false promises about his parents or his true power, making him curious enough to follow them.
Another possibility is that Naruto leaves to protect Konoha. He might discover some kind of impending danger that he believes he can only face away from the village. He doesn't want his presence to put the villagers he cares about in more danger, so he chooses to leave on his own.
In some fanfics, Naruto might leave Konoha as a child because he feels ostracized. Since he's the jinchuriki of the Nine - Tails, many villagers fear and shun him. He could get tired of the constant cold shoulders and decide to leave in search of a place where he'll be accepted.
One reason could be to protect Konoha from a great danger that he can only face by leaving. Maybe there's a powerful enemy who has a vendetta against Naruto but would spare Konoha if he goes far away. Another reason might be a sense of self - exile due to some event that made him feel he no longer belongs in Konoha, like a misunderstanding where the villagers turn against him again.
One possible reason could be a sense of disillusionment. Maybe Naruto discovers some dark secrets within Konoha's leadership that make him question his loyalty. For example, if he finds out that some decisions were made that put the village at risk for the wrong reasons.
One possible reason could be for Naruto's safety. Maybe she has to go on a dangerous mission and believes Konoha is the safest place for him. Another reason might be that she is forced by some powerful enemies who threaten Naruto's life if she doesn't leave him in Konoha.
One reason could be to gain more power. In many fanfictions, Naruto realizes that the threats he faces are far greater than what he can handle with his current skills in Konoha. So, he decides to leave and train in a more secluded and challenging place to enhance his abilities.
One possible reason could be that Naruto discovers some dark secrets about Konoha that shatter his idealized view of the village. Maybe he finds out that the village elders were involved in unethical practices, like using shinobi for immoral experiments. This would make him lose faith in Konoha and decide to leave.
One reason could be a sense of not belonging. In the village, Naruto faced a lot of discrimination because he was a Jinchuriki. As a female Naruto, she might face additional issues like gender discrimination. Maybe she feels that no matter how hard she tries, she will never be fully accepted in Konoha, so she decides to leave.
Naruto might forgive Konoha because he realizes that holding a grudge won't solve anything. He has seen the destruction that hatred can cause, like during the Pain arc. So, he chooses to forgive in order to move forward and bring about positive change. Another reason could be his respect for the previous Hokages and their ideals for Konoha.
They might leave to seek new forms of power or knowledge. There could be things outside of Konoha that they believe will help them become better at their ninja skills. For instance, they might have heard of a special dojo or a hidden scroll in another land that could enhance their abilities.
One possible reason could be to seek stronger opponents outside of Konoha. Naruto might feel that he has outgrown the training opportunities in the village and wants to face greater challenges.