So, what I think is happening is that the victim is necessary to the case in order to prosecute the accused individual. I haven't looked at the video, but it seems from the description that the man does not do anything considered legal assault beyond offensive language and intimidating language. The former is morally repugnant but not a criminal offense whereas the later might be, depending on the context.
In cases where the accused's language and behavior are part of assault charge, then the State must demonstrate that the victim perceived his language and behavior as a threat to the physical well being of the victim (it only counts if you can prove what was said was a threat of imminent lawless action. If the victim did not feel this was the case, then there wasn't an assault or threat of assault). If the victim refuses to testify, then they cannot prove this in a way that the accused's defense cannot pick apart.
In addition to this, and keeping in mind that I have yet to view the video, the physical component of the two punches may have been recorded after the video began capturing the incident. Another issue I found that will be problematic to the case going forward is that while the story says a third party passenger in the car called the BART police to the scene, nobody was able to identify the man despite the fact that he had yet to leave the car when the police left (but was identifiable with the aid of the video). This is not to say that the accused did not do what he is accused of, but more to say that it raises doubt that the story the witness against the man is telling the court is, in fact, the actual events that happened.
In all matters of Criminal Law in the US, the victim has little say regarding what charges are filed. The Prosecutor files the charges on behalf of the legal entity of the state. However, when a victim does say they do not wish to press charges, it means they will not aid the state's case against the accused. Depending on the evidence available to the state and the circumstances of the case, the State might not have the evidence it needs to get the conviction which means no prosecutor will want to go forward with the case. The prosecution will only ever have one shot to convict the guy and a weak case against a guilty person will mean a guilty person will go free.