She may leave due to a prophecy or a vision she has. For example, she might see a future where the village is in danger and leaving is the only way to find a solution. So she ventures out, meets wise hermits or other powerful shinobi who guide her in understanding the prophecy better.
A simple reason could be for adventure. Sakura might be tired of the daily routine in the village and yearn for new experiences outside. So she decides to leave to explore the world and test her skills against new opponents.
Perhaps they leave because they feel restricted in the village. They want to explore the world and grow as ninjas outside the boundaries and rules of the village. They might also want to learn from different ninja cultures and techniques that are not available in their own village.
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.
One reason could be to seek a new life away from the constraints of Konoha. Maybe they feel that they can't fully grow or explore their potential within the village. Another reason might be to protect Konoha in their own way. They could believe that leaving and dealing with threats from the outside would keep the village safer.
She might leave to find herself. In Konoha, she has always been defined by her relationships with Naruto and Sasuke. Leaving gives her the chance to figure out who she is on her own terms, away from the expectations of her friends and the village.
Maybe it's to protect the village. He could discover a threat that he can only face away from the Leaf Village, to keep the danger from directly reaching his home and loved ones. He leaves as a form of self - sacrifice.
In some fanfictions, she might leave to fulfill a prophecy or a special mission. There could be an ancient prophecy that only she could fulfill, and it required her to leave the village at that young age. She might have been guided by some mysterious force or a vision that she believed was leading her towards her destiny.
Perhaps in the fanfiction, Sakura had some unique ability that was not accepted or was feared in the village at such a young age. This led to her family deciding to leave the village so that she could grow up without being ostracized or in danger because of this ability.
Socially, it would create a void. Ino is quite popular and has a lot of friends. Her leaving, along with Sakura's, would leave many people in the village feeling sad and missing them. It could also disrupt some of the social dynamics among the younger generation of ninjas.
Mentally, it could have a big impact on the other villagers. Their friends like Naruto and Sasuke would be affected. Naruto might feel like he failed to keep them in the village. It could also lower the morale of the younger generation who looked up to them as role models.
In the sasunaru context, perhaps a rift between Sasuke and Naruto makes Naruto feel like he needs to leave to sort out his feelings. He might think that staying in the village constantly reminded of Sasuke and their complicated relationship is too much to bear, so he leaves to find his own path and hopefully reconcile with Sasuke on his own terms.