Hi, my degree is in biblical studies and I'm starting seminary this fall. Calling it the "Law" of first Mention is certainly a misnomer. Nobody is getting arrested for breaking this "law."
The principles seems fairly commonsense to me, at least to some extent. If you wanted to learn about Jesus' life you wouldn't start with what people said about him in Acts or the Epistles, you would start in the Gospels. You want to get to the original material as much as possible, because later writers (even biblical authors) will interpret this material in a certain way that might not help you with whatever you're looking for.
This "Law" seems grounded on these principles:
1. The original material is important because it sets the tone.
2. Most concepts in the Bible have a deep and rich context/backstory from earlier parts of the Bible.
3. Failing to go back to the first mentions means you're picking up the discussion halfway through.
4. It is likely that there is some extra importance given to the first mention of a topic in the Bible, intentionally or not.
You could sum it up like this: when you study a particular topic, make sure you take into account the entire context of what you're reading, including how various biblical writers from all parts of the Bible address the issue, and make sure you don't skip over parts just because they don't seem as interesting to you.
Seems fair enough to me, but my feeling is that they try to make it more formal than it should be.
By the way, there are dozens of ways or "schools" of interpreting the Bible. There are a handful of principles that are generally held to be good ideas but it isn't like there is one and only one way to approach the Bible.
-bryan