One possible scenario could be that Bella is out alone at night and a group of drunk guys start harassing her. Edward, with his vampire speed and strength, appears out of nowhere and scares the drunk guys away. He then takes Bella to a safe place, like his house or a quiet café.
Well, Bella is a character who might be a bit naive or just unlucky to find herself in the company of drunk guys. Edward, with his centuries - old wisdom and vampire powers, is the one who swoops in to save the day. He is the protector, the one who will go to great lengths to keep Bella safe. Bella, in turn, is the one who makes Edward feel more human, and his need to protect her is part of what drives their relationship in this fanfiction.
She might be in shock at first. After all, she was in a life - threatening situation. But then she could feel extremely grateful towards Edward and their relationship might grow stronger.
Bella would likely be very frightened during the car incident. Once Edward saves her, she'd be extremely relieved and probably feel even more attached to him. Edward, on the other hand, would be initially worried and focused on getting Bella to safety. After saving her, he might feel a sense of relief as well, but also a bit of that over - protective feeling that he often has for Bella.
In many such fanfictions, Edward might use his superhuman speed to reach Bella just in time. He could have sensed her danger through his vampire instincts and dashed to the cliff edge, grabbing her before she falls.
Edward could use his mind - reading ability. He reads Charlie's intentions and plans his moves accordingly. If Charlie is about to make a decision that Edward believes will be bad for Bella, he can intervene at the right time. He might also be able to manipulate the situation subtly so that Charlie changes his mind without realizing it.
In many Twilight fanfictions, the scenario of Edward saving Bella from Jacob can be quite intense. It often plays on the complex love triangle dynamic. Edward might save Bella from Jacob in a physical altercation, perhaps when Jacob's wolf instincts get out of control. This could be a way to reaffirm Edward's role as Bella's protector in the eyes of the fans who are rooting for their relationship.
One possible reason could be that Edward believes Charlie is a threat to Bella in some way that Charlie doesn't realize himself. Maybe Charlie unknowingly gets involved with some dangerous elements, and Edward, with his vampire senses, detects it.
One key element is the shift in power dynamics. Usually in the original Twilight story, Edward has a lot of control over Bella. In these fanfictions, Jasper disrupts that by saving Bella, which changes the whole dynamic. For example, Bella might become more independent.