How important was Rome's specific location on the Italian peninsula in growing from an unremarkable village into a mighty empire?

To begin with, there are two parts to why Rome is settled where it is- the mythic reasons, and the proven utilitarian ones. According to the myth(s) of Rome’s foundings, first, when Aeneas came to Italy and began to settle down, he was taken on a tour of the ruins at what Rome used to be. In this tour, an “account” of which was given by Vergil, he saw the ruins of a former civilization. Later, in the story of Romulus and Remus, the twins were washed down river as babies and returned there to found a city. While Virgil’s account may suggest Rome was a presettled city, and built off of another civilization, I consider this to be so swiss cheesy that it isn’t worth taking seriously.

On a more factual basis, Rome is founded in a very defensible city in the middle of several important resources. First, Rome is located on the Tiber, which provides a constant source of fresh water. The river also connected Rome to the Mediterranean, which allowed naval trade despite an inland location. Second, it is surrounded by hills, providing a solid defense against attack. Third, Rome is located on a crossroads for the salt trade, or the Via Salaria. The best crossing of the Tiber was located between the Capitoline, Palatine, and Aventine hills in Rome, so Rome connected the salt beds at the base of the river with the rest of the Etrurian civilization. The Etruscans were in Italy before the Romans, and the people who allegedly led Aeneas through Rome were Etruscan, and the Etruscans led the Tarquinian dynasty of Roman kings.

Overall, much like many ancient cities, Rome grew due to its location at the center of trade routes, and with easy access to important resources.

Source:

The Beginnings of Rome: Italy and Rome from the Bronze Age to the Punic Wars by Tim Cornell.

Admittedly, some of the notes on Virgil's story come from lectures with the primary source.