Games teach you how to learn.
Games create simulated worlds, either by showing them to you or through a set of abstract rules and/or physical props. They point you towards a specific goal and encourage you to learn the rules of the game world and skill yourself in the specific way the game requires. Then you use the newly acquired knowledge and skill sets to pass stages, score points, win matches, or otherwise gain a measure of your progression over time.
In other words, education and training.
That’s why games make you smarter. Games don’t just teach you skills, they teach you about the concept of transferable skills; how to recognise different styles of challenge and adapt your existing skill sets to the new situation. Plenty of groups are already creating games that train people in useful skills. The frigging US Army uses a video game to simultaneously attract and train potential recruits.
Games can be both very fun and very powerful tools. The trick I think is that the more we can make game tasks mirror real life tasks (without ruining the fun) the smarter those games will make us.