There is a lot of debate about what a game is, its content and its purpose. Some say that it is a form of art, others say is not. Some say that a game should be fun, but then there are a series of serious games which are not fun but instead try to communicate an idea in an interactive way.
Some say that games are about storytelling or creating imaginary worlds but then there are games like Tetris, and a lot of other board and card games which don't count with stories or any imaginary worlds.
Then there are some who divides traditional games (board games, card games, etc) and video games into different categories so they can define games in terms of content. As for the purpose, it's not clear if games should be done to entertain, develop skills, tell a story, create a world, communicate an idea. It seems the constant in all abstractions of what people take as games is that it is interactive with the player by some mean.
However there's another thing and is the player's purpose on playing a game.
Is different when you talk about a purpose as a developer. Developers want people to have a good time, to tell them a story, show them some very cool audiovisual experiences, maybe teach them about something, or even try to convince them about something. For players it is a lot different.
Games have in all means been taken by players because they really want to. Different from a lot of other things done by people, games are not played because they are supposed to be played or because of a higher purpose (different from studying, exercising, working, marrying, etc). I'll tell you why people play games: they play them for the same reason they do most of the things others never taught them to do, they play because games are a simplification of life.
If you see deep down the reasons why people do most of their leisure activities you'll find that they are really trying to simplify life. Reading is really a way out of the complexity of real life and just as movies they almost always have a happy ending, and when they don't then nothing really happened and they earned the experience of a fictitious sequence of events. Activities such as drawing, painting or writing do the same but actively instead of pasively.
Games are a more powerful tool in terms of simplifying life. The reason is that games not only can provide with fictitious experiences but they can also provide players with fictitious status. Evolutionary psychology tells that one of people's inner desires is to have a respectable social status and this is something that games simplifies in a quite exceptional way.
In other terms, people play games also because they want to compete and this is something that can be seen in the rising trend of social games and how almost all games now have some way to compare a player's performance against other players. If a game doesn't have any mechanism of sharing with other players it doesn't mean that players ain't playing for the sake of competing.
People see status in different ways, some of them see status as who has more, some as who has more friends, some as who achieved the greatest, others as who experienced the most, and some other ways. Games simplify this creating scores, contacts, achievements, and levels. Scores and achievements are characteristics which can be measured easily and summarized in leaderboards so players can compete easily.
Contacts is something that only works for online communities, even social applications such as Facebook show us how some people can worry about gathering the most contacts. Finally levels are the ultimate mechanism for the single player. Some players just like to play with the sole purpose of succeeding in the most number of levels. Passing levels show in a simplifyied manner that they have a better developed skills than others.
Simplification of life is something that attract players for sure. Whether it is to provide an experience, teach something, or put some people to compete and gain status... it is all a simplification of something called life. That's why a game is fun... if instead of simplifying it makes a harder version of life then it won't be as fun.