As you know, Takaki and Yuki actually knew each other since childhood. Back then, Yuki's surname was actually Takashiro. They used to play each other a lot, but Yuki's parents gets divorced, and because she had to follow her mother side, she had to transfer school and drop her surname Takashiro. Yuki gets worried because Takaki might not be able to recognize her with all the changes she will go through. Then well, with Takaki being a kid after all, he offers his surname(…)for her to use if she doesn't like her new surname.

OK this is the most important point of the story cause this pretty much means proposed marriage in Japan (since wife follows the husband's surname). I don't think Takaki was aware of this, but because of that, Yuki was touched and all, and she never forgets this and Takaki after that. Well too bad cause Takaki forgets this from his memory as time goes on (bastard XD)

Back to the present, now named Yuki Kusakabe, she was about to transfer to the high school where Takaki attends. But the day when she was going to transfer (April 24th), she reads the newspaper and finds out that Takaki gets killed in the accident. Refusing to believe, she rushes outside with the newspaper and suddenly got blinded by light (Apprently, she mistakenly ran into the incoming truck) Then suddenly she just found herself lying on the grass and with Takaki looking over her who was at school picking up his half-finished report that he forgot to bring it with him (This is April 14th, and you have to select the first option to go to school). And this is where the story goes on.

So to sum it up, Yuki is a future person. Who is from only a few days on the future. I know, this is really surprising and confusing and all (to quote you: reality bending XD) but this is how the story is. The fact that the game NEVER explains how the f–k she was able to go back a few days of her time (she doesn't know either) or how she came back at the end of the story doesn't help matters either. But since Yuki found herself in the past, I think she tried to prevent the accident that was going to happen in the future, despite not knowing what the f–k just happened, and in the end she ended up saving him when he was almost about to get runned down by the truck. Some hints are shown during their trip to aquarim, when Yuki stops Takaki trying to cross the road when it was still red-light, even though there was no incoming car. She only allows him to go when it turns green light. She was afraid of the accident that will happen.

So in a way, this scenario leaves a lot of questions. I don't know if I was able to explain well (sorry for my messy english) but that's the story there.