Understanding this really does effect how we read our Bibles. There are two schools of thought on this topic: one is that we read the NT in light of the Old; the other is that we should read our OT in light of the New, especially the 'Jesus event'. Which is it? I think it has to be both - which requires multiple readings and rereadings of Scripture. First we read the Old Testament as if we have never heard of Jesus, which requires an active imagination! Then when we get to the new we are stunned at how Jesus steps into the stream of Israel's history. Jesus is like David. He's like Moses. He's like Joshua. He's not like Adam. Like but also unlike Israel herself. He's a better prophet than the prophets. He's a better priest who offers a better sacrifice than the priests. He's the own who ushers in the Day of the Lord with all its terrors and blessings.
Then, having read this, we go back and reread the OT and see all the ways it points us to Jesus. Adam failed to keep the law - someone must come to keep it for us. Melchizedek is a type of Christ, David is too. The tabernacle and especially the temple are types of Christ. The sacrifices prefigure the ultimate sacrifice of Christ. The exodus is a type of the salvation offered by Jesus. So is the return from exile.
Too often we leave off one of these readings. If we forget to read the OT in light of the New, we'll treat the OT as though it's sub Christian literature, like Marcion. If we forget to read the NT in light of the Old we'll neglect the stream of history into which Jesus stepped. We'll neglect the actual importance of events like the exodus and the calling of Abram and the exile and return and end up spiritualizing them or turning them into some wild allegory. We'll miss out on the significance of what it meant for the early church to call Jesus 'Lord' and 'Christ' and 'Son of Man'.
Schleiermacher, the founding father of modern hermeneutics and also the father of liberalism, urged the readers of the Bible to read the parts in light of the whole, the construct the whole from the parts. Know the whole story and read the parts, especially the unclear parts, in light of the whole (also Luther's analogy of faith). When you do this, the ways in which the NT reads and interprets the Old don't seem nearly so weird. On a first read, it doesn't seem like the NT writers always care about the original context or the intent of the author. However, when you see that a passage from Exodus has an immediate context, the verses and chapters immediately surrounding it, and a larger context, the whole sweep of redemptive history, many problems solve themselves.
Did NT authors take things out of context? Maybe out of the immediate context, but not the whole context. They had read the whole story and saw how the parts fit together even better than Moses or David did. Did they NT authors then violate the authors intent in writing a particular passage? If by author you mean Moses or Joshua or the prophet Joel, then you could make a case for that (not one I would agree with, but you could make the case). However, while David and Joel were the human authors, they were only the human authors and the Scripture are not product of only human activity, but of divine revelation. They true author is God - what else could it mean to call the Bible God's Word. The NT authors, while they may appear to violate the original intent of the OT authors DO NOT violate the intent of God, the divine author. Moreover, I don't believe they violate the intent of the human author either. If Moses saw how John used Genesis 1 in the first chapter of his Gospel and knew the whole story of Jesus, I think he'd nod and say, 'yeah, didn't see that in what I wrote, but you're right.'
Thinking through what this means for our hermeneutic is challenging and important. How we interpret Scripture has become an issue of ambivalence to many ('oh that's just your interpretation'), though through history it has often been a matter of life or death. Luther, with his head on the chopping block:
Unless I am convinced by the testimonies of the Holy Scriptures or evident reason (for I believe neither in the Pope nor Councils alone, since it has been established that they have often erred and contradicted themselves), I am bound by the Scriptures adduced by me and my conscience has been taken captive by the Word of God, and I am neither able nor willing to recant, since it is neither safe nor right to act against conscience. God help me. Amen