Much like stepping into the Hyberbolic Time Chamber, you'll be able to send Pokemon down to the touch screen to increase its base stats by participating in mini-games.
For example, one Super Training activity will have you going against Balloon Bots, trying to shoot balls into the opponent's goal by using a combination of the circle pad and touch screen.
But if that's too action-packed, there will be another more casual way of training Pokemon. "Core Training" allows Pokemon to raise stats on their own, by simply tapping the touch screen during a training session.
In an unrelated note, here's Mega Evolution in action: