Although there are many more reasons, John Stott does a good job summing up the main reason I believe God is the ultimate author of the Bible: because Jesus believed it.
To accept the authority of the Bible is a Christian thing to do. It is neither a religious eccentricity, nor a case of discreditable obscurantism, but the good sense of Christian faith and humility. It is essentially “Christian” because it is what Christ himself requires of us. The traditional view of Scripture (that is God’s Word written) may be called the “Christian” view precisely because it is Christ’s view.
The ultimate issue in the question of authority concerns the Lordship of Christ. “You call me ‘Teacher’ and ‘Lord’,” he said, ‘and rightly so, for that is what I am’ (John 13:13). If Jesus Christ is truly our teacher and our Lord, we are under both his instruction and his authority. We must therefore bring our mind into subjection to him as our Lord. We have no liberty to disagree with him or to disobey him. So we bow to the authority of Scripture because we bow to the authority of Christ. (Taken from John Stott: The Making of a Leader, p. 356).
Among the many reasons why Christians believe the Bible is God’s very Word, top among them is because Jesus did.