The Plot Explained (SPOILERS)
Wow, what a movie! Many people on this board seem to be a tad confused, so here's a brief summary that should help elucidate the story. It’s mostly a walkthrough of the events presented during the montage. This post is one big spoiler, so stop reading if you haven’t seen the movie yet.
The main thing to understand is that the real protagonist of the story is the Yakuza with the sunglasses. The film starts with him (I'll call him Yakuza) and Atsu waiting in the car. When they see Myon run by, Atsu gets out and chases her. She jumps into the train, and Atsu manages to get onto the same train so that he can follow her. This ends the first sequence.
Next, the montage of past events begins. This is a chronological run through of all of the events that lead up to the scene in the car that we just witnessed. I'm going to go by the reprise of the montage that comes at the end of the film, because it is more complete.
The first shot is of a bridge. This is the first memory of the baby in the 2nd shot, who will grow up to become the old man from the whale. We see him reach adulthood, where he chooses to lead a life of crime. (Metaphor can be seen in the church shot) We see him have a child (who grows up to become the Yakuza) and we see that he smuggles drugs in children's toys. He is then swallowed by the whale after a drug bust. His wife subsequently dies during the childbirth of Atsu.
We now see his first child, the yakuza, who has grown up. He meets a woman and they fall in love. This woman will become the mother of Myon and Yan. One night at the club, she sees Myon and Yan's father and chooses to go with him, essentially dumping the yakuza. This is what propels him towards becoming a yakuza. (it's unclear what his profession was prior to this) The yakuza spirals into a life of crime, and the woman gives birth to Yan and Myon. However, the father continues his womanizing ways, and pays little attention to his family. Nishi is also born.
Now the focus shifts to the childhood of Myon, Yan, and Nishi. Also interspersed is Atsu's lonely life, where we see him become a soccer player. He eventually fails at that endeavor. Subsequently, he meets the crime boss who recognizes him from his soccer exploits and hires him. We also see shots of Yan, Myon, and their mother cowering in a dark room, ignoring the ringing phone. Their father has racked up debts that they cannot pay. Ironically, those debts were loans from the crime bosses’ syndicate. We see Atsu, who now works for the yakuza with his brother, trying to collect on the loans.
Meanwhile, Myon’s mother is secretly meeting with the yakuza, since Myon’s father has essentially abandoned them. The yakuza now drives the car that the father used to drive, so we can assume they repossessed it as payment towards a portion of his debt. Eventually, Myon’s family moves in order to escape the bill collectors. They open up the Yakitori shop, and Myon’s father comes back to live with them since he has no other place to go. This is the point at which Nishi loses contact with Myon.
However, Myon’s mother maintains her relationship with the yakuza. He eventually gives her a train ticket to Tokyo, and tells her that the two of them should run away from all of their problems in Osaka. At the same time, Myon’s father steals Atsu’s girlfriend from him. She also takes his world cup tickets, which is why he is enraged and wants to track him down. This brings the viewer up to speed on all of the past events that lead to the first scene in the car.
The film then goes into the story of Nishi, Myon, and Yan. What the viewer needs to understand is that the first scene (Atsu chasing Myon onto the train) and the entire Nishi story that follows is the yakuza’s dream. He wakes up from his dream at the end of the film, and decides to run away with Myon’s mother rather than pursue his (and Atsu’s) revenge against Myon’s father. This is why “this story has never ended.” It is the beginning of a story, rather than the end. Really, the message should read “This story has not ended.” I would chalk the error up to poor grammar.
The reason that the yakuza has a change of heart is open to interpretation, but I believe that he simply considers the dream to be an epiphany. Nishi decides to actively choose how to live his life in the dream, and I think that the yakuza wishes to do the same. Rather than continue down his bloody path of crime, he elects to escape it all with the woman he loves. The dream serves as both a warning and an awakening for the yakuza. He realizes all of the possibilities that life has to offer, characterized by the “future” montage which we see right after Nishi jumps out of the whale’s mouth and into the sky. I think that the yakuza envisions Nishi as himself within the dream, but that’s a whole other post.
This became a lot longer that I was anticipating, but I think that it’s still relatively easy to understand. Hope this helps!