Paul talked a lot about Abraham's justification by faith (as recorded way back in Genesis) throughout Romans 4. But Paul noted that "it was not written for his sake alone that [righteousness] was imputed to him" (4:23). Basically, although the many Bible stories we heard are stories that happened in the past, the themes of these stories can still hold true today. After all, part of the reason why we study history is learning from it.
As for the connection to today, just like how Abraham was justified by faith, we too are justified by faith -- faith in "Him who raised up Jesus our Lord from the dead" (4:24). In particular, Jesus "was delivered up because of our offenses", and Jesus "was raised because of our justification" (4:25). The resurrection demonstrates that Jesus has power over death -- the ultimate side effect of sin due to separation from God (as seen back in Romans 1). So in a sense, Jesus (along with the great exchange) can prove that he can counter death and offered himself to us (who, if we're honest, have done various screwups). Also, from the perspective of Hebrew law, Jesus represents the perfect sacrifice done to cover our sins.