What does the rule that "Scripture interprets Scripture" mean and where does it fit into a larger picture of the field of hermeneutics? What hermeneutical approaches use this as a guiding principle?
It simply means that the scriptures must harmonize. The orthodox Christian view of the Bible is that it is not in error and does not contradict itself. Therefore when trying to examine one passage we must approach it with an eye to what the whole Bible says about that topic. For example, let's take divorce.
While this is certainly clear from a plain reading, this is a controversial topic. You wouldn't see that from here, would you? When we look at what the Bible says about divorce in other places you can see why there is a dispute within the church on the topic.
This example is good because we can see a place where Jesus actually gives commentary and builds upon the OT. The key to this idea of "scripture interpreting scripture" is that you cannot look at one passage or verse in a vacuum. To earnestly understand what the Bible (and therefore God) says on a subject we must look at the whole of the book.
Sola Scriptura/Regula Fidei
The idea behind Sola Scriptura is that the Bible is complete and sufficient for all knowledge of salvation and holiness. From this doctrine, the Bible should be used for judging faith and soundness of doctrine and practices; it is, therefore, the "Rule of Faith", or measure by which we judge something.
Scripture interprets Scripture
From this idea that scripture alone was sufficient came the idea that we should interpret scripture using other scripture. Since the Bible alone was the sole measure for judging faith and practice, it should be used as a measure to understand other parts of scripture.
As a Hermeneutic Approach
As a hermeneutic approach, "Scripture interprets Scripture" is the idea that we should read a passage in the light of the entire Bible. It also states that we should interpret confusing passages based on clear passages.
"Scripture interprets Scripture" is, in itself, a hermeneutic approach. However, it's also a mind-set used by other hermeneutic approaches.
Other Hermeneutic Approaches
This idea that scripture can be used to gain insight into other passages in the Bible is found through many hermeneutic approaches.
Sensus Plenior is one example that uses "Scripture interprets Scripture" as a general guiding rule. Theological Analysis is a technique that shows that theology must harmonize throughout all the scriptures (so that one small passage doesn't create doctrine).
The idea that we can use scripture to interpret other scripture is a common theme throughout Hermeneutics. From the perspective of Sola Scriptura, it is, by itself, a hermeneutic principle.
Ultimately, it is simply the idea that we can use scripture to shed light on other passages within the Bible.
In sensus plenior it means that all the answers to the riddles are found within scripture.
Jesus says we should let our light shine before men, yet we should not let our left hand know what our right hand is doing. There are at least twelve different interpretations of this "apparent contradiction" by people using the literal-historic methods.
Jesus says Goats are on the left and sheep are on the right. So we should do the works of God for which we were created, but we should not let our 'goat nature', the flesh, interfere with the spiritual works we do. We should not be claiming credit for God's work as Nebuchadnezzar did.
So the Ninevites who did not know the left from the right, did not know the flesh from the spirit.
The sword-word (in Eglun and Ehud) was pulled from the right (spiritual) thigh-will with the left-flesh hand-works. God's spiritual will was accomplished by men working in the flesh.