They are popular because the Dursleys are such complex and disliked characters in the original series. Fanfictions give readers a chance to explore their motives and potentially redeem them. For example, we can understand why they treated Harry so badly in some fanfictions which might be due to their fear of the unknown world of magic.
Maybe it was an accident. Like a wrong Floo Powder destination or some kind of misfired spell that landed Draco right at the Dursleys' doorstep just as Harry was there.
It could be because the authors want to emphasize Harry's difficult upbringing and show how he overcomes adversity. Also, it makes the contrast with the magical world more pronounced.
Because many fans feel that Harry was mistreated by the Dursleys in the original story. So they create fanfictions where he is taken away from that bad situation. It gives Harry a chance at a different and hopefully better life.
Maybe Draco finally understands that Harry is a victim of circumstances. The Dursleys' treatment of Harry is cruel, and Draco, having grown up in the wizarding world, realizes it's wrong. So he steps in to rescue Harry.
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.
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.
Well, the Dursleys were really horrible to Harry. Fans want to see different ways he could have been saved. It also allows for new storylines. For example, if he's saved early, how would his magical development be different? How would his relationships with other characters change? All these possibilities make these fanfics interesting. Also, it gives a chance to explore different magical elements and characters in the context of rescuing Harry from the Dursleys.
Well, in the original story, the Dursleys are really mean to Harry. So, fans want to explore more about Harry's feelings towards them. It gives an opportunity to write about his pent - up anger and frustration.