Why does self-injury make some people feel better?

There are a few possibilities, and the answer is probably a mixture of them. Biological predisposition, reduction of tension, and lack of experience in dealing with strong emotions are all factors.

It reduces physiological and psychological tension rapidly.
Studies have suggested that when people who self-injure get emotionally overwhelmed, an act of self-harm brings their levels of psychological and physiological tension and arousal back to a bearable baseline level almost immediately. In other words, they feel a strong uncomfortable emotion, don't know how to handle it (indeed, often do not have a name for it), and know that hurting themselves will reduce the emotional discomfort extremely quickly. They may still feel bad (or not), but they don't have that panicky jittery trapped feeling; it's a calm bad feeling.

This explains why self-injury can be so addictive: It works. When you have a quick, easy way to make the bad stuff go away for a while, why would you want to go through the hard work of finding other ways to cope? Eventually, though, the negative consequences add up, and people do seek help.

Some people never get a chance to learn how to cope effectively
We aren't born knowing how to express and cope with our emotions -- we learn from our parents, our siblings, our friends, schoolteachers, -- everyone in our lives. One factor common to most people who self-injure, whether they were abused or not, is invalidation. They were taught at an early age that their interpretations of and feelings about the things around them were bad and wrong. They learned that certain feelings weren't allowed. In abusive homes, they may have been severely punished for expressing certain thoughts and feelings. At the same time, they had no good role models for coping. You can't learn to cope effectively with distress unless you grow up around people who are coping effectively with distress. How could you learn to cook if you'd never seen anyone work in a kitchen?

Although a history of abuse is common among self-injurers, not everyone who self-injures was abused. Sometimes, invalidation and lack of role models for coping are enough, especially if the person's brain chemistry has already primed them for choosing this sort of coping.

Problems with neurotransmitters may play a role
Just as it's suspected that the way the brain uses serotonin may play a role in depression, so scientists think that problems in the serotonin system may predispose some people to self-injury by making them tend to be more aggressive and impulsive than most people. This tendency toward impulsive aggression, combined with a belief that their feelings are bad or wrong, can lead to the aggression being turned on the self. Of course, once this happens, the person harming himself learns that self-injury reduces his level of distress, and the cycle begins. Some researchers theorize that a desire to release endorphins, the body's natural painkillers, is involved.