In some fanfictions, it could be due to a special jutsu gone wrong. Maybe a jutsu that Kakashi was using to transfer some of his chakra to Naruto for a mission accidentally transferred his Sharingan power as well.
Well, Naruto might gain new abilities. He could potentially use the Sharingan's powers like predicting movements better in combat. It would be a huge boost to his already formidable skills.
One possible reason could be that Kakashi is in a life - threatening situation and realizes that Naruto is the best candidate to inherit his Sharingan. He knows Naruto's potential and his desire to protect the village. Another reason could be that Kakashi wants to see how Naruto would use the Sharingan in combination with his own unique abilities. He's curious about the new powers and techniques that Naruto could create with it. Also, it could be a way for Kakashi to show his complete trust in Naruto, as the Sharingan is a very precious and powerful gift.
One great fanfiction could be about Kakashi using his Sharingan to train Naruto in a special jutsu. Maybe he shows Naruto how to better perceive an opponent's moves by using the Sharingan's ability to analyze. This could lead to Naruto developing a new and powerful technique that combines his own chakra control with the insights from the Sharingan.
His powers would change a great deal. Naruto is known for his brute strength and his ability to never give up. The Sharingan would add a whole new layer of finesse to his fighting style. He could use the Sharingan's genjutsu capabilities to trap his enemies in illusions before hitting them with a powerful chakra - infused punch. Also, he could use it to analyze his opponents' weaknesses more quickly, allowing him to adapt his fighting strategy on the fly.
As for his relationship with the rest of the village, some might be suspicious at first. The Sharingan has been associated with the Uchiha clan, which has had some dark moments. But as Naruto shows that he can use the power for good, they would gradually accept it. However, there would always be some who are wary, which could create some interesting social dynamics in the village.
His relationship with Sasuke would be greatly affected. Sasuke might see Naruto as more of an equal in terms of their eye powers. They could have a new level of understanding or even rivalry. It might also make Sasuke more protective of Naruto in a strange way, since the Sharingan is precious to the Uchiha.
For his relationship with Kakashi, it could be complex. Kakashi, having his own experiences with the Sharingan, would be in a unique position to guide Naruto. But at the same time, he might be worried that Naruto's new power could attract unwanted attention. Naruto's bond with the villagers could also be affected. Some might see him as more of an Uchiha than a Uzumaki now, and there could be those who fear his power. His relationship with Jiraiya, if he were still alive in the fanfic, would also change. Jiraiya would need to teach Naruto how to use this power for good and not let it corrupt him.
His relationship with Sasuke would become more complex. Sasuke, who has always been proud of his Uchiha heritage and the Sharingan, might feel a mix of envy and pride that Naruto now has a part of what makes the Uchiha special. Naruto, on the other hand, would try to use the Sharingan to understand Sasuke better, perhaps seeing the world more from Sasuke's perspective.
Naruto's relationship with his friends in the Hidden Leaf Village would also change. Some might be in awe of his new power, while others might be a bit afraid. For example, Sakura might be worried that the Mangekyou Sharingan could change Naruto's personality. Naruto would have to reassure them that he's still the same person. And he might also use his new power to protect them more effectively, strengthening their bond.