One reason could be that he discovers a new threat and decides he can't stay with the Dursleys any longer for their safety. Another might be that he simply can't bear their mistreatment anymore and finds an opportunity to escape.
One possible plot development could be Harry finding a new magical family or community that accepts him wholeheartedly. Maybe he discovers a hidden magical society that has been observing him at the Dursleys and now offers him a real home and proper magical training.
In some fanfics, when Harry leaves the Dursleys, he might immediately feel a sense of freedom. He could start exploring the wizarding world more independently, without the constraints of his Muggle relatives. Maybe he'd meet new friends or allies earlier than in the original story.
There could be other young witches or wizards who are also on the run from something or have their own issues with the wizarding establishment. They bond with Harry over their shared experiences and together they face new challenges. Maybe a young witch who has unique powers related to nature and she helps Harry understand the more elemental aspects of magic.
In some versions of the fanfic, Harry might have a sort of instinctive understanding that his parents had a good reason for leaving him there. So, while he's still not happy about it, he might be a bit more accepting. Maybe he can sense some sort of protection spell around the Dursleys' house that gives him a small sense of security.
The plot could be that Harry's parents are in a magically-induced coma. The Dursleys are tasked with looking after Harry until a cure can be found. Harry might feel a sense of isolation as he is with the Dursleys but constantly thinking about his parents. He could start to show early signs of magic, which terrifies the Dursleys even more, but also gives him hope that he can somehow use his powers to wake his parents up.
In the 'the dursleys abuse harry fanfiction', the abuse can be multi - faceted. Physically, Harry could be pushed around or hit by Vernon. Emotionally, Petunia might constantly compare him unfavorably to Dudley, which is really damaging to Harry's self - esteem. And socially, they might prevent him from having any friends over, making him feel completely alone in the world.
It's possible. In some fanfictions, authors might explore complex motives for the Dursleys' actions. Maybe they could be under some dark magic influence that makes them act out of character and in a more extreme way. However, this goes against the general perception of the Dursleys in the original series, where they are mainly just cruel and neglectful without such extreme murderous intent.
One consequence could be that Harry becomes withdrawn. He's so used to the abuse that he doesn't trust people easily. He might have nightmares often, reliving the moments of abuse. For instance, if Vernon has hit him before, he could have flashbacks in his sleep.