It could be his own past experiences. Maybe he once lost someone to suicide or has faced a similar dark moment in his life, and he doesn't want the girl to go through the same pain. So he is determined to save her.
His sense of justice. Power Rangers are all about doing what's right, and saving a life in danger is a top priority for them.
In the fanfiction, it could be that the Power Ranger has some sort of psychic connection to the girl. Maybe he was in the area and his heroic instincts kicked in. He could then use his communication skills, which are part of his heroic toolkit, to talk the girl out of it. He might share his own stories of facing difficulties and overcoming them, making the girl realize that there is hope.
One possible motivation is a sense of morality that Draco has buried deep inside. Despite his upbringing in a family of Death Eaters, he still has a conscience. Seeing Greyback about to harm Hermione, who is essentially an innocent in the face of such a brutal werewolf, stirs that conscience. Another factor could be his own self - interest. He might realize that by saving Hermione, he could gain some sort of leverage or protection in the complex world of the wizarding war.
Love. In fanfictions, it's a common motivation. Draco might have fallen in love with Hermione over time, and that love drives him to risk his life to save her from Voldemort.
It could be out of loyalty to the wizarding world. Harry is seen as an important figure in the fight against the dark forces. Fleur realizes that Harry's safety is crucial for the future of the wizarding community, so she saves him from the Dursleys who are a threat to his well - being in a different way compared to the dark wizards.
Stiles is a very loyal friend. He would see Erica and Boyd in trouble and feel compelled to help. In the show, he has always been there for his friends, and this would carry over into the fanfiction. He can't stand by while Gerard does something bad to them.
It could be a combination of factors like extreme stress, unfulfilled desires, or a sense of hopelessness.
Snape might be motivated by a desire for redemption. He has a complicated past, and saving Harry could be his way of making up for all the bad things he has done or been associated with. Additionally, he might see a bit of himself in Harry's situation, as he also had a difficult childhood, and this empathy drives him to act.
In many fanfictions, Kakashi might save Naruto from suicide by using his Sharingan. He could see Naruto's inner turmoil and use his jutsu to snap Naruto out of his dark thoughts. For example, he might use a genjutsu to show Naruto all the people who care about him and how much he means to the village.
Maybe she's trying to get their attention. In a world full of mutants like in TMNT, it can be hard to be noticed. The girl might think that this rather bold action of pulling on their ears will make the mutants focus on her. It could be that she has a message for them or wants to ask for their help, but doesn't know any other way to approach them.
One possible motivation could be guilt. Snape has always had a complicated relationship with Harry. Maybe he feels guilty for not preventing Harry from being wrongly accused and sent to Azkaban. So, he decides to make it right by rescuing him.