Maybe he's tired of the spotlight in England. In some fanfictions, he could be seeking a new start away from the prying eyes and the constant media attention that comes with being in England.
One reason could be to escape the trauma of the war. They've been through so much in England, and leaving gives them a chance to heal in a new environment.
One reason could be to escape the danger and dark forces that still lurk in England even after the main battle. They might feel that leaving is the only way to start anew and be truly safe. Maybe they want to explore other magical cultures around the world and learn more about magic that is different from what they've known in England.
One reason could be to explore the wizarding world outside of England. There are many magical places they might be interested in, like the wizarding communities in France or America. They could be on a mission to learn more about different magical cultures and bring back new knowledge to England.
In some fanfictions, Harry might leave his family because of a prophecy or a special mission. He might feel that he has to go it alone in order to fulfill this prophecy or complete the mission successfully. His family might not understand or be able to support him in this, so he decides to leave. Also, there could be jealousy or resentment within the family that makes the environment unbearable for Harry.
In some cases, he could be leaving in search of a greater purpose. He might feel that there are things he can learn or achieve outside the wizarding world that will ultimately help the wizarding community more. He may also leave because of some prophetic vision or a feeling that he doesn't belong there anymore, that his path lies elsewhere, away from the familiar magical realm he grew up in.
One reason could be a falling - out with his friends in Gryffindor. Maybe there was a misunderstanding that grew so large that he felt he couldn't stay. Another possibility is that he might be lured away by some mysterious force or character who convinces him that Gryffindor is no longer the right place for him. For example, if a new group claims to have a better way to fight Voldemort and they are from a different 'house' in this fanfiction concept.
One reason could be that he's tired of the constant danger and politics in the magical world. He might want a normal life away from all the dark wizards and power struggles.
One reason could be to find peace. After all the chaos and war in England, they might just want to get away from it all and start fresh somewhere else.
One possible reason could be that in the fanfiction, the wizarding community in England has turned against Harry due to some false accusations. Maybe a powerful dark wizard has framed him for a series of heinous crimes, and the public, being misled, forces him to leave. Another reason might be political intrigue within the Ministry of Magic. There could be a new regime that sees Harry as a threat to their power and thus kicks him out.
Another reason could be a misunderstanding. Maybe he overhears something Harry and Hermione say and misinterprets it as them not wanting him around anymore. So, in a fit of anger and hurt, he leaves. This can lead to a lot of angst and drama in the fanfiction as the other characters then have to figure out how to get him back.