Learning how to wrap your hands properly for boxing can be one of the most frustrating things when starting out, leading many to question why it is necessary. Wrapping your hands properly is crucial to your health and safety when training in boxing for two main reasons:
- Hygiene - if you don't bring your own gloves and pads for your boxing workout, then having your hands (and everyone else's) wrapped when training is a critical extra layer of hygienic protection, to help prevent the transmission of not just things like the common cold, but more serious issues like staph infections, and just prevents that "yuck" factor of sweaty gear, as the hand wraps will absorb the sweat instead of it getting into the lining of the pads or gloves. Even if you DO have your own gear, having your hands wrapped each time you use your gloves or pads can go a long way to keeping that gear fresh and funk-free almost indefinitely for the same reason.
- Protection - wrapping your hands properly can help protect your wrists and hands from damage due to the repeated impacts involved in boxing training, especially as you become more advanced and can hit with more power. You can vary the way your hands are wrapped depending on where your hands or wrists need more structure or protection - I.E. if your wrists are particularly unstable, you can use more wrap across the wrist joint to protect it.
One note regarding the second point above - if you are training simply for fitness purposes, or are training for sport/competition where your hands will be wrapped, then absolutely, you can rely on wrapping your hands for the purposes of structure/protection.
If, however, you are training for the purposes of self-defence, or the potential of actual combat WITHOUT hand wraps or other protection, remember that you will not be able to hit as hard un-wrapped without injuring yourself. Your best compromise when training in this scenario is to try and distribute the hand wrap as evenly across your hand as possible, and with minimal extra wraps at the wrist and across the knuckles. For this type of training, I also prefer to use as minimal a glove as possible so that I get a better feel for how hard I can actually hit without hurting my own hands.
Here's a great video on how to "wrap your hands to increase your punching power and prevent hand injury" from expertboxing.com.
Practice, practice, practice!