What does games like The Sims, World of Warcraft, Bioshock, Dragon Age, Braid, Final Fantasy, and even Tetris have in common? They are all successful games and they all tell the story about the hero who challenged all the puzzles, adventures, enemies and setbacks of the game. It attracts players because it satisfies a psychological need to succeed whether it is represented on a personal character development or an ingame character development.
You can't appreciate the benefits of character development on most of all successful games (there are a few exceptions off course) and I believe this is probably the most profound communication that there is between player and game. There's a lot of research to be done yet about the psychological implications but there are a few guesses that can be made about why people play games: 1) a distraction, 2) a desire to develop a (gaming) skill, or 3) a curiosity about what the game is trying to say.
Games as a distraction may be interpreted as games as a way for the player to escape from the routines of real life. Games which better produce the distraction desired for a player are those games that allows the player to transfer his perception of the world into the game. This is probably true because the sensation of entirely loosing time is not sympathetic with the player's desire so there must be some kind of intrinsic link between the game and the player's character. A good example about this are games such as The Sims (the player relates with the characters in the game), World of Warcraft (there's a relationship with real people competition), or facebook games (the same "real" competition relationship as in WoW).
A DESIRE TO DEVELOP A SKILL
Some games relate directly with the internal desire of a player's character development. These games are just saying: "if you play me long enough you will develop skills which will be useful for your real life". The best example of this are the Brain Age games, however any puzzle game can be cataloged under this type of character development. Puzzles games just say: "if you play me you'll look smarter". I write "look" because puzzle games are somehow deceptive: they don't make you smarter, they just make you feel smarter and that's why you play them (with the exception that puzzle games could be disctractions to the player or even trying to attract the player for its "message").
A lot of arcade games also fit this. For example Tetris, everytime somebody achieves a higher level it is saying: "now you are a faster player, your reflexes are better in real life". I also read once somewhere that heavy Tetris players had better concentration and attention skills in real life.
A CURIOSITY ABOUT WHAT THE GAME IS TRYING TO SAY
At last there are some games which communicate with the emotional side of the players. Just like reading a book or watching a movie, some games attract players by sharing an idea, insight or a story. This type of games are probably the favorite for players who like to learn by experience. They like where the games lead them because is like discovering the world at a faster pace (and without losing, you can always retry). For games trying to call on the player's curiosity they must make a lot of work on ingame character development since these must relate deeply with the player.
In the end, there's a lot about character development (not story character development) in games. There are a lot of other games trying to communicate with players in other ways and achieve unconventional reactions on its users. When designing a game, probably the best way of proceeding is to achieve the three reactions on players (distraction, desire of skill development, and curiosity about message) since they are all compatible between each other.