With so many different way to look at the world, be as simple as good or evil in fable.. fighting demon in diablo.. being an alien race encountering humans.. Or the simple watching someone cook some eggs.. it all helps you to think outside the box.. and to open your mind to other possibilities. You can't just play a game one way because sometimes your idea doesn't work, and other times (as AI gets better) they react to that strategy making it futile over time. You have to think about multiple scenarios in quick amount of time. It helps you to not only make decisions on the go but also to visualize your goal while being timed.... Like say setting up that quintimple play in Dr. Mario... you don't have the room for a mistake, nor the time to think that long.. yet you need the play to make your friend want to kill you.
Games are beyond button mashing and simple fun, but now they take your mind to get past puzzles, they take quick decision making to keep on living, some games take precise movements in the physical world to dodge bullets (a definate work out on the upper thighs I might add). Where some give you a physical work out, other put your mind to a great test. Some levels seem like a dead end until you realize that if you shoot the air duct and jump off that box over there that you can get inside and move to the next level. All of these are still able to keep the game fun.
There are also games meant to be educational in purpose.. they put pieces of fun here and there while learning useful skills.
You can learn to catch a beat with games like DDR or the new Donky Konga game (essential for any would be musician). You can draw and create pictures for any would be artists. You can strain your mind from tough puzzles for any would be scholar. You can think outside the box for any would be lawyer. YOu can also get into shape for any would be Athlete.
And aside from all this.. its a lot of fun!