When you first play the game, those video montage offers a glimpse of the story that lies ahead. As you progress through the game, the montage turns into a mixture of scenes you have seen and those you have not. These montages are carefully mixed so there are no big spoilers but just enough teasers to entice you to play on and on. I loved picking out the scenes to come and predicting how the story will unfold based on those scenes.
It really does feel like a TV serial and in a lot of ways, the Yakuza series is actually more like a TV serial than a summer blockbuster. The cutscenes are usually grounded on human drama with most of the action occurring during the game play. That's probably one of the reason why the series have difficulty breaking out of Japan in terms of mainstream success. For all its criminal undertones, it's more about the characters themselves rather than the shady stuff that yakuzas do.
Casual observers often compared the Yakuza games with Grand Theft Auto which is such a poor comparisons. You hardly ever do any criminal activity in the Yakuza games, just regular badass stuff that makes you feel like the boss.
Badass stuff like this following video montage