Harry may save Draco from Buckbeak simply because he is a good person at heart. He has always been against unnecessary harm, and seeing Draco in danger from Buckbeak, his instincts would be to help. It could also be that the author wants to explore a different side of their relationship, where Harry's compassion overcomes their rivalry, and this event becomes a turning point in how they interact with each other in the fanfic world.
Well, perhaps Harry uses his quick thinking and his knowledge of magical creatures. He might step in just in time, using a spell to distract Buckbeak before it can harm Draco. For example, he could use a simple levitation spell to lift a small object and draw Buckbeak's attention away from Draco.
Well, perhaps in this fanfic, Harry notices Draco in a dangerous situation with Buckbeak. Maybe Draco was taunting Buckbeak and it got out of hand. Harry, being the brave one, steps in and uses his skills with magical creatures to calm Buckbeak down just in time to save Draco from getting seriously hurt.
For Harry, his friends in Gryffindor could be a bit confused at first. They know Draco is not exactly a friendly character. But overall, it could lead to a more united front against real threats in Hogwarts. The teachers might also take notice. Dumbledore, for instance, would probably be proud of Harry for showing such selfless courage, regardless of who he was saving.
Well, it could be for power. Maybe Voldemort realizes that by saving Harry, he can gain some sort of power over him or use him as a tool in a more elaborate plan.
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.
It's popular because Buckbeak is such a cool and unique character. He's a hippogriff, which is a very interesting magical creature.
In this fanfiction, Harry might also save Draco by appealing to Lucius' sense of self - preservation or family. He could threaten to expose some of Lucius' secrets or illegal activities if he doesn't let Draco go. Or, if there is some underlying family drama involved, Harry could use that to his advantage. For instance, if there's a feud within the Malfoy family that Harry is aware of, he could manipulate the situation in a way that makes Lucius back off and leave Draco unharmed.
Well, it could be that Draco has a change of heart early on. He realizes that the Dursleys are truly awful to Harry. So he uses magic to create distractions. He could make strange things happen around the house that scare the Dursleys so much that they just want Harry to leave. Then Draco could be waiting outside and whisk Harry away to a safer place, like a hidden place in the wizarding world where he has set up a little haven for Harry.
Maybe Draco discovers a hidden weakness of Voldemort. Draco, having been raised in a pure - blood family, might have access to some ancient magic knowledge. He uses this knowledge to create a situation where Voldemort is temporarily incapacitated, giving Harry the chance to escape or defeat Voldemort. For example, he could find a way to disrupt Voldemort's connection to his Horcruxes for a short time.
Well, Harry might save Daphne from Draco because Harry is always the hero type. He doesn't like Draco's pure - blood supremacist attitude. Daphne could be a target of Draco's cruel remarks or actions just because she might not fully conform to Draco's ideas. Harry, being the defender of the underdog in many cases, steps in. Also, it could be that the plot of the fanfiction is building a new alliance or relationship web, and Harry saving Daphne is a key part of that, leading to new story developments.