One reason could be that they see Loki as a victim of circumstances. Maybe they've come to understand that his actions were often influenced by forces beyond his control, like Thanos' manipulations. So they decide to defend him in Asgard.
Maybe Loki is wrongly accused of a crime in Asgard. The Avengers, having had their own experiences with false accusations, step in to defend him. They believe in justice and know that Loki might not be guilty.
One possible reason could be his feeling of being an outcast. In the original stories, Loki is often treated differently in Asgard due to his nature. He might have felt like he didn't truly belong, which could fuel his hatred. Another reason could be envy. Maybe he saw others in Asgard having things he desired, like power or recognition, and his envy turned into hatred.
He could meet the Frost Giants. Since he has some connection to them already, they might welcome him or at least be intrigued by his presence. He could learn more about his heritage from them and maybe even form an alliance.
One reason could be for the sake of creativity. By making Loki a kid in fanfiction, it gives a fresh perspective. Writers might want to rewrite his origin story in a more innocent - starting - point way. Also, it can add an element of cuteness or sympathy to his character that may not be as prominent when he is depicted as an adult. For example, the Avengers might be more hesitant to fight a child Loki, which can lead to interesting plot twists.
They show him kindness and try to understand his motives. They offer support and help him find a better path.
One possible plot could be Loki leaving Asgard after feeling constantly overshadowed by Thor. He might leave in search of a place where his skills and intelligence are truly valued, perhaps finding a hidden realm where he can start anew and build his own following without the judgment of the Asgardian court.
In fanfiction, the Avengers might learn about Loki's hidden motives. For example, some fanfictions depict Loki having a more complex backstory regarding his actions in the MCU. He could be revealed to have been acting out of a sense of misunderstood loyalty or a desire for acceptance that was misdirected.
I also like 'Asgard's Love for Loki'. In this fanfiction, Odin, despite his past mistakes regarding Loki, decides to be more protective. He uses his power and influence to shield Loki from external threats and also tries to mend their broken relationship. It's an interesting take on how Asgard can step up to protect one of its own who has often been seen as an outcast.
In the fanfiction, Loki discovers he is pregnant just as the plans to build new Asgard are underway. He is initially worried about how the Asgardians will react. Thor, being his brother, stands by his side and assures him that new Asgard will be a place of acceptance. As Loki's pregnancy progresses, he uses his magic to help in the construction, creating beautiful illusions that inspire the builders. In the end, when the new Asgard is complete, Loki gives birth to a child who is seen as a new hope for the future of Asgard.
In fanfiction, if the Avengers go to Asgard, there could be a big clash at first. The Asgardians are proud warriors and the Avengers have their own ways. But then they might team up to face a common threat. For example, a powerful dark magic user could be targeting both Earth and Asgard. The Avengers with their diverse skills and the Asgardians with their magic and strength could form an unbeatable alliance.