Punching faster means your strikes land earlier, more often and with greater force. It should come as no surprise, then, that many fighters make punching faster a priority in their training. Holding hand weights while punching adds resistance to your punch, which can build attributes that improve your speed, though not as effectively as one might hope.
Strong muscles can accelerate faster and move objects like your fists with greater speed. This is why sprinters spend time in the weight room. If you punch while holding hand weights or dumbbells, you place extra load on your arm muscles. This can make your muscles stronger and better able to accelerate your arms and hands into the punches.
Although it may look different from the outside, a properly-thrown punch involves much more than your arms. It starts with your foot digging into the ground, and engages your legs, rear, torso and shoulders before flowing out through your arm. Hand weights put most of their load on your forearms and shoulders, meaning that they engage only a fraction of the muscles involved in a punch.
Muscle strength can contribute to punching speed but is not nearly as important as solid technique, relaxation in the moment or tactical considerations. A martial artist or boxer learns dozens of techniques and skills that build up to fast, accurate punches. Physical conditioning is just a small part of that development.
Walking or otherwise exercising with hand weights is very hard on the shoulders, according to Dr. Mehmet Oz in "You: Losing Weight." It can turn a simple workout into increased potential for repetitive stress injuries due to the pull on your shoulders and upper arms. This risk is exacerbated while punching because your extended arm places maximum stress on that area.
While it is true that punching with weights can build muscles that are partially responsible for punching speed, that fact is not the whole story. Punching with weights engages only the tiniest fraction of the factors that make punches fast and increases your risk for injury during training. It is not recommended as a speed-building practice.