When you prove something, you show that it's true. If you say you love eating raw eggs, you may have to prove it by chugging a few. When someone asks you to prove something, you need evidence, also known as proof.
Prove comes from the Latin root probare, "to test or prove worthy." The word’s meaning is clear in this quote from writer Ernest Holmes: "There was a time when a man was so convinced that the world was round that he was determined to prove it." As a transitive verb, prove requires an object to receive the action, as in "How can you prove your claim?" It can also be intransitive, as in "You might prove to be right."