Most important, games teach cause and effect to children. They teach that nothing comes without brainwork, that there are rules, that reality to be commanded must be obeyed. They teach children how to cope with not getting the right answer right off the bat, an essential emotional skill. Games teach children the joy of getting better at something, as well, the necessary positive reinforcement that reassures them that losing isn't so bad and is a temporary state of things which they have the power to change. They teach that even if you get dealt a bad hand, smart play over time can pay off.
Games also provide a safe arena for all human virtues and vices to be tested and played out.
Teamwork in games teaches trust and leadership, even through all the temporary strife of unsuccessful attempts at teamwork, trust and leadership which inevitably accompanies the playing of games. The most marvelous and unique quality of games is that the short lifespan of a game makes sure that loss and victory, success and failure can happen quickly and thus get the whole picture of life across within a single sitting. The fact that games are a microcosm of life allows the experience to be digested whole, and in the right balance. A child who plays no games might have a crushing experience, an awful try-out for choir or a dismal failure to give a speech, and have only that failure to dwell on until the next significant moment comes along. The child who plays games knows that setbacks are a temporary part of life and that there's always a next time when, with some effort, one can have more success.