Carmine, the Rosato brothers were a breakaway faction in Frank Pentangeli's family (actually the remnants of the Corleone Family's New York operation). They were at war with Frankie because, as Michael told Frankie in the boathouse during Anthony's party, "Clemenza promised the Rosato Brothers three territories in the Bronx after he died. He died, you didn't give it them." The meeting you described was suggested by Michael as a way for Frankie to "settle these problems with the Rosato brothers," but was in reality an attempted murder ordered by Roth. The guy Frankie met outside the bar was Carmine Rosato. The guy who attempted to garrotte him inside was Tony Rosato.
BTW: The Rosato brothers probably are modeled on the real-life Gallo brothers (Larry, Crazy Joe, Albert), who broke away from the Profaci family (later the Magliocco Family, later the Columbo Family) around 1960. Larry was garrotted in the back of a bar in Brooklyn, but the attempt was thwarted by a cop who happened by--the model for the scene you were wondering about.