To outdo is to beat or surpass: when you outdo your brother in a pie-eating contest, you manage to gobble down more lemon meringue than he does. You'll probably regret it later, but at least you'll have that blue ribbon!
Anyone who outdoes another person wins against them in some contest, whether it's an actual competition or an unstated rivalry. You can also outdo your own record, like when you outdo last semester's grades with the help of all the As you get in English. And you can cheer for your favorite basketball team as they outdo their opponents by easily winning the game.