It is important to note that chapters were added to the biblical text by later copyists and editors for ease of navigation and citation, not by the original authors themselves. The chapter break between 1 and 2 is artificial, for 2 continues the train of thought found in 1.
As Paul develops his narrative that mankind has descended into sin, he establishes that sin is universal. All humans are tainted by it, both Jew and Gentile.
In establishing that sin is universal, Paul can then develop this point to show that all humans need salvation if they are to survive the judgment of God. This idea Paul then focusses on Israel to expose hypocrisy.
He attacks the idea that one does not need salvation if one adheres to the Law of Moses, the Torah. It is important to understand the nuance here. Paul is not saying that the Torah is of no value. Nor is he saying that a person who keeps the Torah perfectly need fear the judgment of God.
Rather, he is saying that one must keep the Torah perfectly in order to withstand the judgment. But since no one can keep the Torah perfectly, as is evident from the sins of Israelites, one cannot be saved by attempting to keep the Torah.
This will become clearer in subsequent chapters, where Paul will discuss justification in more depth. At this stage, he is establishing that (1) Israelites are in need of salvation, just like the Gentiles, and (2) Israelites should be humble before God because of point (1), especially in light of the fact that the Gentiles themselves obey the moral precepts of the Torah (albeit imperfectly).
Paul is writing to a mixed audience, and he is trying to build a community where there is neither Jew nor Gentile in Christ, but all love one another as Christ loved us. (Although, as we shall see later, Paul’s doctrine of Israel is complex.)
Paul reminds his readers of the eschatological consequences of sin, for those who do not receive salvation: judgment during the Day of the LORD and in the hereafter. This note of warning and sobriety is another way of impressing upon the audience its need for salvation.