Maybe it's because of Harry's status as a Triwizard champion. Just like her, he's in the spotlight during the tournament. Fleur might think that being with another champion would enhance her own standing. Also, Harry has a certain charm about him that might have caught her eye during their interactions related to the tournament.
It could be that Fleur has noticed Harry's kindness. Harry is known for being kind - to his friends, to those in need. In the wizarding world full of various personalities, this kindness could stand out to Fleur. And she, perhaps, wants to get to know him better by asking him out.
One possible reason could be that Fleur sees Harry's courage. In the 'Golbery of Fire' events, Harry shows great bravery, and Fleur might be attracted to that. She herself is a confident and somewhat proud character, and a brave partner like Harry could be appealing to her.
Harry could be flattered. Fleur is considered very beautiful, and for her to show interest in him would boost his self - esteem. He might stammer out a response, something like 'I - I'm honored, Fleur.' while blushing furiously.
In the fanfic, perhaps there will be a lot of blushing and nervousness from Harry. Maybe he's initially taken aback by Gilbert's forwardness. He might need some time to think about his feelings, as he has so much else going on in his life, like dealing with Voldemort and his Hogwarts studies.
One possible reason could be a misunderstanding. Maybe Fleur misinterprets Hermione's actions as a threat to something or someone she cares about. So, she restrains Hermione to prevent her from causing further 'harm' until she can figure out what's really going on.
Gilbert might be an original character created specifically for this fanfic. He could be a student at Hogwarts from the Fire Fleur house, which is likely a fictional house made up for the story. He may have unique magical abilities or a distinct personality that draws him to Harry.
In some fanfictions, it could be due to a prophecy or a magical bond that they discover. They might feel that the only way to fulfill the prophecy or strengthen the bond is by getting married secretly. Another aspect could be the political situation in the wizarding world. There could be political factions that would oppose their union for various reasons, such as Harry being seen as a symbol of a certain side and Fleur being from a different background. So, to avoid political turmoil or to keep their relationship safe from political manipulation, they choose to marry secretly.
In one plotline, Fleur could be the target of a curse that slowly drains her magic. Harry, as her bodyguard, has to find a way to break the curse while keeping her safe from those who might want to take advantage of her weakened state. This could involve seeking out ancient spells and potions, and they might have to journey to dangerous and mysterious places. As they do so, their relationship could evolve from Harry simply being a protector to someone Fleur deeply trusts and relies on, and there could be hints of a romantic connection developing between them.
Some writers might do it for shock value. They want to create a story that stands out from the typical fanfics.
Perhaps Fleur is being controlled or influenced by a dark force. In the Harry Potter universe, there are many dark magic entities that could manipulate someone into kidnapping Harry. It could be that she is under the Imperius Curse or some other form of mind control. Another reason could be that she has some sort of personal vendetta against Harry or those close to him, and kidnapping him is her way of getting revenge. She might think that Harry has wronged her in some way, whether it's real or just in her deluded perception.
One possible reason could be Fleur's unique charm. She is part - Veela, which gives her an alluring quality that might make Harry feel insecure. Another reason might be her skills. If in the fanfiction she is shown to be extremely talented in magic, especially in areas where Harry struggles, it could trigger his jealousy.
Fleur might expose them. If they are students, they could face expulsion from Hogwarts.