I've had this in draft mode for over two months, waiting to put some finishing touches on it. But since this is not a formal thing, I decided I'd just go ahead and post what I have. You'll find a few similarities here to my recent "2,500 years" and "430 years" posts. After all, it's the same subject matter.
"Now we know that whatever the law says, it says to those who are under the law..." (Rom 3:19a, emphasis mine).
I have known this verse (and the entire surrounding section of scripture) for a long time, but a few years ago as I was reading it, the above emphasized part stood out to me like never before. Whatever the law says, it says to those who are under the law. Lights went on in my head and I pretty much spent the next six to twelve months camped out on those words, and I began to understand so many other passages of scripture that had previously confused me. I eventually came around to the second part of the verse, which I'll get to shortly because it's very important in understanding the "why" of the first part, but it was that first part that really began to illuminate so many things for me. Please follow me all the way through.
If the law has anything to say - and it has a LOT to say - it is speaking to those who are under the law (not to those who aren't under it). This has always been true. It didn't become true just when Paul wrote those words. He was simply revealing a truth to the Romans (and eventually us) that was already true. It was true when the law was given. It was true when the prophets (and any and all other Old Testament characters) spoke the words of the law. It was even true when Jesus spoke the words of the law. Whatever the law says, it says to those who are under the law.
Remember, Jesus said, "Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill" (Matt 5:17). He then went on to say, "For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled" (Matt 5:18).
Paul tells us in Gal 4:4-5, "But when the fullness of the time had come, God sent forth His Son, born of a woman, born under the law, to redeem those who were under the law, that we might receive the adoption as sons" (emphasis mine).
We see at least two things here. 1) Jesus, born under the law, came to fulfill the law. 2) Jesus came to redeem those who were under the law.
Which actually brings us to the second half of the first verse mentioned here. Here's the full verse:
"Now we know that whatever the law says, it says to those who are under the law, that every mouth may be stopped, and all the world may become guilty before God" (Rom 3:19, emphasis mine).
In the full verse we see two things that the Law was given to accomplish. 1) The law came to stop every mouth. 2) The law came to make the world guilty before God.
So whatever the law says, it says to those who are under it. Why? (Or for what purpose?) To stop the mouths (of those who are under it) from justifying themselves in front of God and to make them guilty before God. (Rom 5:13 says that sin was in the world before the law, but the law was needed in order to actually impute sin - to make people guilty by putting sin on their account).
Addendum (a few hours after original posting):
In the comments, the question came up of what "make the world guilty before God" means. I thought this was important, and I realized that I had mentioned Romans 5:13 in the above paragraph, but hadn't gone into any detail about the word "impute," so here's what I said in the comments:
That key word, "imputed," makes all the difference. It means to add to one's account. Sin was in the world, but the world had not yet been actually charged with sin. It had not been put on their account. When the law came, man was actually charged with sin, and "made guilty before God." This was necessary, in order for Jesus to come and take the guilt of the world upon Himself.
We find the law all over the Bible, in both the Old Testament books and in the New Testament books. But no matter where we find the law, we must keep in mind what Paul said: Whatever the law says, it says to those who are under it. And whatever the law says to those who are under it, it says for the purpose of stopping their mouths and making them guilty. The law aids in no way in helping a person to be righteous nor to redeem them or give them life.