When you disbelieve someone, you're unable to trust what they're saying. A detective is likely to disbelieve a suspect who's sweating nervously and not making eye contact.
Believing something or someone means you've got confidence or faith. Add the prefix dis- to believe, and it's the exact opposite. If your cousin is famous for his tall tales, you'll probably disbelieve his story about meeting the Queen in England last summer. And if a jury disbelieves someone's testimony, they're likely to discount it when they start deliberating.