I do not practice boxing or martial arts, but I am a tennis player, and every coach will say the same about hitting the ball as a martial arts coach will about hitting an opponent - the power should come from your core and lower body, rather than your arm/shoulder. The idea is to use large muscle groups (quads, hamstrings, glutes, core) to create movement. These are the strongest muscles in your body, so they will allow you to hit with not only more power, but more stability as well. If all your power comes from your arm and shoulder, it will be easy for someone to deflect the hand or stop it altogether, as they will be fighting rather small and (relatively) weak muscles.
The physical mechanism is simple - using your legs and core, you can force your upper body to rotate. If your body is rotating, then one shoulder is moving towards your opponent. Hold out your hand and use your arm/shoulder muscles to guide it as you rotate, and you will have a more powerful punch than if you kept your shoulders stationary and punched with your arm muscles.
I do not have any training suggestions, other than practicing with this in mind. It helps to keep a slight bend in the knees so that your quads are already engaged. Your legs should be tired at the end of a practice session. As far as conditioning goes, I would suggest rotation exercises with a medicine ball (holding the ball in both hands, rotate it from one hip to the other). To step it up, do it on an upside-down bosu ball.