Maybe he wants to escape the constant danger in the UK wizarding world. Voldemort's followers might still be a threat even after his defeat, and France could seem like a safe haven.
One possible reason could be that he feels unsafe at Hogwarts. Maybe there are threats from dark wizards within the school that the teachers are unable to fully protect him from. Sirius, being his godfather, represents a safe haven.
In fanfic, one reason could be that he might feel constantly in danger at Hogwarts. With all the dark forces targeting him, he could think running away is the only way to keep his friends safe. For example, if Voldemort's followers keep infiltrating the school, Harry might feel responsible and decide to leave to draw the danger away from others.
Perhaps in the fanfiction, Harry Potter runs away from Britain because he is disillusioned with the way the Ministry of Magic operates. There could have been a major event where the Ministry fails him or betrays his trust. Also, he may be seeking a place where he can explore his magic without the constraints and expectations that come with being 'The Boy Who Lived' in Britain.
Perhaps he runs away because he feels misunderstood. In the fanfiction, the other students or even some of the teachers might wrongly accuse him of something. He could be tired of constantly having to defend himself. Like if he was wrongly blamed for a serious incident at Hogwarts, and no one seemed to believe his innocence, he might just leave in frustration.
They might run away because they want to explore the wizarding world on their own terms. Without the restrictions of Hogwarts or the watchful eyes of the Ministry of Magic.
His friends would be extremely worried. Hermione, for instance, would likely try to find him using all her knowledge and resources. She'd be scouring every book and magical spell she knows to track him down. Ron would also be concerned and probably want to go after him right away, but he might be a bit more hesitant about how to actually do it.
Snape would likely be angry at first. Harry has always been a source of trouble in his eyes, and running away would just add to that perception. He might mutter some sarcastic remarks under his breath about Potter's irresponsibility.
Snape might suspect that there is some dark magic at play. Given his knowledge and expertise in the Dark Arts, he would immediately look into any signs of foul play. He could also be conflicted because on one hand, Harry is a student he has had many issues with, but on the other hand, he is also an important figure in the fight against Voldemort. So he would have to decide whether to report Harry's disappearance or try to handle it privately.