"I picked up the dice and threw two sixes. Caldecott couldn't believe it. My go again; another two sixes! Anyway I picked up the dice again... Unbelievable! Another two sixes! Then -- disaster! I threw a two and a three; Caldecott picked up the dice and threw snake eyes -- I was still in it. Anyway, to cut a long story short I threw a five and a four which beat his three and a two, another double six followed by a double four and a double five. After he'd thrown a three and a two I threw a six and a three."
My point is that some games lend themselves to interesting narratives more easily than others. Of course the narrator makes a difference, there are people who can weave a compelling narrative around a game of Risk and those who could reduce Tales of the Arabian Nights to a sequence of mechanical effects, but the nature of the game makes a difference. The importance of a game's narrative to players will vary from person to person, some people won't play a game without a strong narrative while others don't notice if one is absent entirely (There are also players who will find a way to weave a narrative into even the most abstract of games). So while acknowledging that it's not a big deal to everyone let's get into why it matters to a great many players.
There are a lot of advantages to a strong narrative, but for me there are three that stand above the others. It makes rules more comprehensible, changes how you feel about the game and changes the stories that are told about the game. I've talked about the relationship between mechanics and narrative before so won't go into detail here. In a nutshell people can learn complex rules more quickly when they relate to and are similar to concepts that are already understood.
Using narratives to change how people feel about a game can also be pretty important. For most great old ones the dominant strategy in Arkham Horror seems to be to spend the game tooling up for a fight and defeating it when it finally awakens, but few people fall into this approach during their first game. Even once the approach is discovered a lot of players won't approach the game that way. The reasons people give are things like "Exploring the outer realms is interesting." or "Monsters on the streets killing everyone is awful." Ultimately these reasons stem from the narrative, mechanically the outer realm cards are no more interesting than the Arkham location cards, they cause you to gain and lose the same things. The game uses the narrative to draw players into approaching it in the intended way, which is just as well since it's often a more enjoyable game in that respect.
Besides changing how people play the game, there are some games in which the narrative is the point. The aforementioned Tales of the Arabian Nights barely exists as a mechanical entity, it's much more about telling an interesting story than winning a game. The influence that the players exert on the mechanical outcome of situations that they are in is minimal; in reality their influence principally impacts the story. When you choose to punch the sailor you push the narrative in a particular direction, but whether it turns out that you righteously dispatched a traitor or struck a genie in disguise is not up to you so it's hard to have a meaningful impact on the actual outcome of the game. Games in which telling a good story is more important than (and irrelevant to) winning play very differently to more traditional games, but in a way that adds variety to the sorts of enjoyment people can get from games. I'll always prefer a crunchier game, but I have to acknowledge that I can't remember how to play Gloom while still holding onto the memory that I was menaced by midgets.
The third narrative benefit I mentioned applies after the game is finished. People get excited about games and it's very human to want to describe things that touch you to others. Where a concept is hard to grasp and put into words this is a frustrating experience. Having played for years with groups of gamers that play in separate times and places but meet up once a week and swap stories I've heard more than my fair share of gaming anecdotes and some games suit them better than others. The stories of the head of vecna or sir bearington can engage people who've not played those games and maybe even tempt them to join in. Space station 13, a computer game with some board game sensibilities (that should really get a post of its own someday, as it expresses the best and worst of game design) has been described as "an anecdote generator".
I wouldn't say that the nature of stories produced from gaming sessions is the main reason to have a narrative focus, but I think it does add something to a lot of people's lives.
It's good for players to be able to share stories. It's good for people to enjoy the stories that are shared.
It's good for games to be played by new people lured in by these stories.
Which is good for the people playing the game and the game's creators and everyone else involved.
Tell me a story.