After nearly a decade of apostle-ing churches, having founded at least ten to that point, Paul had developed a pretty good working idea of what kinds of problems young churches would deal with - especially problems between Jewish and Gentile believers in those churches. But five years after the Emperor Claudius kicked all the Jews out of Rome, he was poisoned and died. Hearing that news, late in 54 AD, Paul was finally able to put Italy into his own travel plans. Postponing his impending trips to Corinth and Judea, Paul went as far as Dyrrachium, founding a church there as a mid-way rest stop for the future, and for any stragglers among his few dozen friends who were moving to Rome (see Rom.16).
Twenty-nine months (or so) after Claudius died, Paul still seemed no closer to visiting Rome, with his hands full in Corinth/Cenchrea. However, Paul knew many other Jewish-Italian believers (converted at or since Pentecost, like Priscilla & Aquilla) would have moved back to Rome, by that time. Feeling responsible for the 30 to 40 (or more) saints he'd
That's my 2 paragraph explanation as to why Paul wrote Romans.
Now I'll add two paragraphs of commentary.
That story doesn't only explain why Paul wrote Romans, it also explains why Romans came out the way it did. Paul wasn't trying to be systematic. He was just trying to communicate well to a large group with very diverse perspectives, in order to help mediate between close friends and virtual enemies. Paul was trying to phrase things helpfully for some who may have been more familiar with James' epistle & ministry, and for others who likely preferred the language of Galatians.
But in all that, Paul's motive wasn't to craft some ideal theoretical treatise for unity. He was engaging very practical concerns, among both groups, trying to challenge and honor both groups, and his motive at that time was simply to build common ground among these particular Jews and Gentiles whose particular conflicts and impasses had in fact - to that point - prevented them from standing together as one body, one church.