One possible reason could be a misunderstanding. Maybe Naruto overheard something his family said that he misinterpreted as them not loving or believing in him. For example, if he heard them talking about his wild nature or his past mistakes in a way that he took the wrong way.
However, it could also have negative impacts. He might start to feel lonely and bitter, which could lead to him making some bad decisions. He could be more easily influenced by the wrong people or ideas because he lacks the grounding that his family usually provides. Also, his relationships with his friends might become strained as he tries to keep them at a distance too, thinking that they'll side with his family.
One possible reason could be the burden of his past. Naruto has faced a lot of hardships and the memories might be overwhelming him, making him want to be alone. For example, the loss of his master Jiraiya might have left a deep scar, and he might feel that isolating himself is the only way to deal with that pain.
His relationship with the village would also change. The villagers, who have come to rely on Naruto as a hero, would be confused and perhaps a bit scared. Some might feel betrayed that he has isolated himself. But there could also be those who understand his need for solitude and wait patiently for his return.
Perhaps he's dealing with some sort of trauma. For example, after the many battles he's been through, like the fight against Voldemort, he may be suffering from post - traumatic stress disorder. To cope with this, he isolates himself from others as he doesn't want to burden them or he's just too caught up in his own dark thoughts.
In the later part of the plot, there could be a turning point. Maybe Harry realizes that his isolation isn't helping him and that he needs his friends. Or perhaps an external threat arises that forces him to come out of his self - imposed isolation. This could lead to a rekindling of his relationships, and they all work together to face the new challenge stronger than ever before.
Well, in fanfictions, often the plot takes a different turn. Naruto might hide from his family because there could be some sort of misunderstanding. For example, there could be false rumors about him being involved in something bad. And he might think that his family would believe those rumors without giving him a chance to explain. So, to avoid the immediate confrontation and judgment, he chooses to hide.
Naruto could be afraid of being seen as weak. In the ninja world, weakness is often frowned upon. He has a reputation to uphold, especially in front of his family. He doesn't want them to think that he can't handle his problems, so he hides his pain. He also might be trying to be strong for his family. If they see him in pain, it could affect their morale. So, he pretends that everything is okay to keep their spirits high and to give them the impression that they can rely on him no matter what.
This is a really dark concept. In fanfiction, such a scenario might be written to explore extreme emotional distress. Maybe the author wants to show how much pressure Naruto has endured secretly, and on his birthday, which is supposed to be a happy day, it all becomes too much to bear. However, in the original Naruto story, Naruto is a character full of hope and perseverance, so this is a very different take.