How did ancient Rome's location and transformation as a republic help begin its rise as a civilization?

The creation of the Roman Republic did not help to begin Rome's rise as a civilisation. Rome had already risen as a civilisation during the previous monarchic period.
The foundation of Rome refers to the creation if the Roman city-state. It involved the unification of separate settlements on what where to become the Seven Hills of Rome (the Palatine, Caelian, Esquiline, Viminal, Quirinal, Capitoline and Aventine) into a single state under the rule of a king.

In those days people liked living on hills because they were easier to defend from raids. The hills and the valleys between them offered ample space for urban growth. The volcanic rocks of the area gave good building material. Rome was on the only ford on the river Tiber, which made Rome the main north-south communication centre in the region. The Tiber also afforded navigation to the sea, which was only 14 miles away (16 nowadays). When Rome developed a thriving trade network around the Mediterranean, the Romans built a port on the coast (Ostia) by the mouth of the river. Large ships unloaded here. Goods were then transferred to smaller ships which sailed to Rome by sailing up the river. Being inland also meant that there would not be any naval attacks. Rome was also not far from the Apennines (a mountain chain which runs through the Italian peninsula). These mountains had plenty of water sources. When Rome's demand for water exceeded the supply form local sources, water was taken to Rome from the mountains by the aqueducts.

At a wider level, when Rome expanded and developed an empire around the Mediterranean Sea (during the period of the Roman Republic) she benefitted from Italy's strategic location. Italy lies in between the western and eastern basins of the Mediterranean. This gave Rome a good strategic position to turn both westwards and eastwards and establish control over both the western and the eastern Mediterranean