To testify is to make a statement or provide evidence, usually in court. Witnesses testify for the prosecution or defense.
If you've ever seen a TV show about lawyers, you've probably seen someone testifying: putting their hand on the Bible, sitting in the witness seat, and telling what they know about a case. Testifying is serious business, but it doesn't always happen in court: If someone knows you're good at math, they can testify to your math skills. Or if you've been to a restaurant, you can testify to your friends that it has awesome desserts.