Via the excellent Brainpickings, this animated video by ASAPScience takes just two minutes to explain how music acts on the brain to trigger a sensation of pleasure.
In short? By targeting the brain's reward systems and stimulating the release of the neurotransmitter dopamine. That's the exact same neurochemical process that takes place when you have sex, eat amazing food, or take drugs (or even simply desire those things). Key quote?
In the same way that a drug-induced dopamine surge leaves you craving more, music becomes addictive — the dopamine tells your body it was rewarded and creates a desire to seek out more.