To corroborate is to back someone else’s story. If you swear to your teacher that you didn't throw the spitball, and your friends corroborate your story by promising that you were concentrating on math homework, she might actually believe you.
For example, a witness in court corroborates the testimony of others, and further experimentation can corroborate a scientific theory. Near synonyms are substantiate and confirm. Corroborate, originally meaning "to support or strengthen," was borrowed from Latin corrōborāre, formed from the prefix cor- "completely" plus rōborāre "to strengthen" (from rōbur "strength").