In Romans, Paul moves from his introduction (1:1-12) to the explanation for not coming to see the Roman churches earlier (1:13-15) to his thesis statement (1:16-17) to his supporting argument (1:18-15:13). Paul's argument in support of his thesis statement is, so far, (1) that all non-Jews are without excuse for obeying God and are therefore under condemnation (1:18-32) and (2) all Jews who know the law are also without excuse and, therefore, are also under condemnation (2:1-3:8) and, in fact, (3) all people everywhere are without excuse and condemned to an eternity in hell (3:9-20).
Paul then moves to argue that only through faith (belief in-and trust in-) in King Jesus is any person rescued from alienation from God. Paul then asserts that, in fact, it has been this way in redemptive history before, specifically, in regards to Abraham. In chapter 4, Paul sets forth Abraham as an example of someone justified (i.e., reconciled to God) by faith (belief and trust) alone.
Because Paul sees faith itself as a gift (we will get to election in Romans 9)there is, therefore, no place for boasting or bragging within the body (3:27 and 4:2).
As a side note, as one of James Dunn's former students, Simon Gathercole, has argued, this section is one of the toughest for folks like N.T. Wright and other adherents to the New Perspective on Paul to deal with because, in their view, one is welcomed into the covenant community by grace but maintain that status by works, would this not lead to boasting? Tim Keller has also noted that such a view could easily fall into the same legalism decried by Paul.