One possible reason could be that Obito sees Naruto as a means to fulfill his own twisted ideals. Maybe he wants to shape Naruto into a weapon against the Hidden Leaf. Another reason might be that he feels some sort of connection to Naruto because of their shared experiences of being alone and ostracized in the past. He could also think that by taking Naruto as a baby, he can prevent Naruto from becoming the hero the village wants him to be, thus disrupting the balance in the ninja world.
The story could develop with Obito training Naruto in his own way, away from the values of the Hidden Leaf. Naruto might grow up with a different set of skills and a different view of the ninja world. There could be a lot of internal conflict in Naruto as he discovers the truth about his origins and the fact that he was taken from his village. This could lead to a showdown between Naruto and the Hidden Leaf at some point, or Naruto might try to find a middle ground between Obito's teachings and his own sense of right and wrong.
It might be for the greater good. If Obito's actions in the fanfic are leading to destruction on a large scale, Konan, being a protector of sorts, decides to take him out to save the world or at least her part of it.
In a fanfiction, Obito might be sent on a mission by some unknown force to watch over Baby Naruto. As he does, he starts to question his own beliefs. Naruto's pure and unwavering will even as a baby might start to chip away at Obito's hardened heart. This could lead to Obito trying to influence Naruto's growth in a positive way, like teaching him some skills secretly without Naruto knowing it was Obito.
Besides Obito, Sasuke could be a main character in this fanfiction. Sasuke, with his own journey of redemption and his understanding of power and the ninja world, could have an interesting dynamic with the reincarnated Obito. Also, Sakura might play a role. She could be the one who tries to heal the reincarnated Obito, both physically if he is injured during his new adventures and emotionally from the traumas of his past life.
Another plot could be that Naruto is tricked into believing a false narrative about the ninja world. Similar to how Obito was manipulated by Madara's lies. He then starts to act in ways that are contrary to his usual self. He might target the other villages, thinking he is doing the right thing for some grand, but misconstrued reason. This would lead to him being at odds with his former friends, who would be desperate to bring the old Naruto back.
One possible consequence is that Naruto would gain enhanced visual abilities. The Sharingan can see chakra flow clearly, which would be a great asset in battle for Naruto. He could better analyze his opponents' moves and weaknesses.
In this fanfiction, Obito's development can be quite interesting. He may begin with his own motives that are not entirely pure. However, as he spends more time with Naruto, he starts to see the world through Naruto's eyes. This could lead to him becoming more empathetic towards others in the ninja world. For example, he might start to understand the importance of protecting the village and its people, just like Naruto does.
Maybe in the fanfiction, Obito makes a deal with an enemy that Naruto sees as a betrayal. So Naruto, with his strong sense of loyalty to the village and his friends, decides to abandon Obito. This could lead to a series of events where Obito tries to redeem himself in Naruto's eyes.
One possible motive could be to use Sakura as a bargaining chip. If he has some sort of plan against Naruto or the Hidden Leaf Village, Sakura, being a close comrade of Naruto, could be very useful. He might think that by restraining her, he can force Naruto to come to a certain location or make concessions.
Obito might be initially hesitant and a bit awkward around Baby Naruto. But as time passes, he could develop a soft spot for the baby. He might play simple games with Naruto like peek - a - boo or make funny faces to make him laugh.