Why do I feel the need to physically harm myself when I feel depressed?

At first, it can be a form of punishment, or a stress relieve. Self harming, or cutting, causes pain in your body. Your brain will release chemicals like endorphins, to deal with this pain, and make it tolerable. Over time, your body can become addicted to these chemicals, and then you feel a need or craving for them. Your body associates these chemicals with self harm, so it's negatively reinforcing the idea that self harming makes you feel good and better.

It can help you return to reality in way. When you self harm it may make you feel like you can be back in reality. It can also possibly be because you feel numb and it helps you to feel like things are real.

Emotional and physical pain are dealt with in the same areas of the brain. When some people feel emotional pain (depression, anxiety, anhedonia, other such pain), they seek to relieve this pain through physical pain. It sounds counterintuitive but with things like cutting/burning, the physical pain is sharp, acute and intense and then the removal of that pain feels like "relief" - as in, once you stop. This "relief" can help relieve the emotional pain. However, self harm is not physically or emotionally safe and is only a temporary solution to ongoing emotional issues that will come back again and again.