What does the Bible say about its own authority?

The Bible says,

• No prophecy was ever thought up by the prophet himself.
  

  But know this first of all, that no prophecy of Scripture is a matter of one's own interpretation, for no prophecy was ever made by an act of human will, but men moved by the Holy Spirit spoke from God (2 Peter 1:20-21).

• It is given by God to give us wisdom to accept God's salvation and help us to do good to everyone
  

  ... and that from childhood you have known the sacred writings which are able to give you the wisdom that leads to salvation through faith which is in Christ Jesus.

  All Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness; so that the man of God may be adequate, equipped for every good work (2 Timothy 3:15-17).

• The prophets themselves did not fully understand what the Spirit of God within them was talking about
  

  As to this salvation, the prophets who prophesied of the grace that would come to you made careful searches and inquiries, seeking to know what person or time the Spirit of Christ within them was indicating as He predicted the sufferings of Christ and the glories to follow (1 Peter 1:10-11).

• God has actually given them the Holy Spirit to explain the Holy Spirit's facts
  

  For who among men knows the thoughts of a man except the spirit of the man which is in him? Even so the thoughts of God no one knows except the Spirit of God.

  Now we have received, not the spirit of the world, but the Spirit who is from God, so that we may know the things freely given to us by God, which things we also speak, not in words taught by human wisdom, but in those taught by the Spirit, combining spiritual thoughts with spiritual words (1 Corinthians 2:11-13).

Also Fulfilled prophecy proves that the Bible is God's Word.