How can painkillers become addictive?

Effects on brain: The opiates work centrally in the brain. They effect not only parts of the brain to control pain but also the pleasure parts of the brain. Not everyone who uses painkillers become addicted.

Chemical dependency: Most people who use opioid based painkillers never get "addicted". Addiction is the psychological need for the feeling a drug gives you. Most people who take pain killers become physically and chemically dependent on them, which is not an addiction. But since these meds stimulate certain areas of the brain that can cause "euphoria", true addiction is possible, but rare.

Brain: Pain meds act on the reward center of your brain which make you want to keep having that feeling. It can also led to accommodation which means your going to need higher does in the future to get the same effects.