1) G. K. Beale explains why this issue is central to understanding your New Testament: the prevalence of citations, allusions, and echoes throughout the NT to the OT. He also explains some sources you can use to find them.
2) To get a sense for why this topic matters, consider a certain text. In Rom 3:9-18, Paul cites six Old Testament proof-texts to support his claim that all are under sin, Jew and Gentile alike. As John Piper explains,
That seems plain enough. But here’s a problem that troubles me. In that same Psalm which Paul quotes to say, “There is none righteous, not even one” it says (in Psalm 14:5), “God is with the generation of the righteous.” In other words, the first verses of the Psalm are a description of the Gentiles who oppose the true people of God, but those who are the true people of God, David calls “the generation of the righteous” (see also verse 7). So the question is: How does this Psalm support Paul’s point in verse 9 that says, “Both Jews and Greeks are all under sin”? The part of the Psalm that he quotes doesn’t seem to mean everybody is unrighteous, because verse 5 calls God’s people “the generation of the righteous.”
I think the answer goes something like this:….
Although you could just listen to his explanation (it starts at 11:55), I recommend listening to the entire sermon to get the full impact of how the study of the NT use of the OT is both theologically and pastorally important.