I'm trying to understand what's wrong with my work.
The instructor said that the strength of the punch comes from the body and from footwork, and not from the arm itself.
The body - picture the human body with a rod through it, from head to the floor. The body can pivot around that rod, but is rooted to the spot. When a punch is delivered, the body pivots. This is where the force is generated. The jab particularly is a weak punch if thrown using the arm muscles alone. But holding your hands head-high, imagining the rod through your head and you're forced to pivot on it, now throw the jab, turning your shoulders so that they follow the leading fist. You get power and extension. A jab thrown with arm muscle only generates power only at the extreme end; if intercepted along the way, the leverage is poor and the strength is weak. If the shoulders rotate, the mechanical advantage is always present, from start to finish. When you hit pads instead of a bag, you're hitting just short of the ultimate finish, so when throwing muscle-only, you lose power compared to punching a bag, where you measure your punch to be at max extension when it lands.
The stance - for each action, there is an equal and opposite reaction. In other words, when you push forward (punch), you are also pushing your body backwards an equal amount. Your body is more massive than your fist/arm, so you may not notice it as such, but it's there. Your body rocks back as you throw your punch forward. However, by rooting your stance so that the body cannot easily move backwards, you increase the force going forward without increasing effort - in other words, a harder punch. If that's hard to imagine, try to imagine yourself suspended in air and punching. Every time you connect, you push your own body back and forth in the air, don't you? That's what happens to a lesser extent when you're not firmly planted on the ground.
Pivoting and generating power from the shoulders, hips, and legs also helps to counteract the force that otherwise pushes you back as you punch forward, because you're balancing the force out with the force in.
Consider the monkey drum:
http://www.magictails.com/abydos/images/mid-east/percussion/large/monkey.jpg
You spin it in the palm of your hand, and it generates force enough to cause the balls and the end of the strings to beat the drum. The string clearly has no power to punch the drum - the strength is generated by the rotation. This was demonstrated in one of the Karate Kid movies, sort of.