How did the government of Rome develop into an empire

Rome was initially a city-state with about 50 square miles of territory to sustain itself. With small farms, the citizens could not split them up between their sons and so sought extra land for them. The surrounding city-states were bent on the same need, so they clashed, with Romme losing some, but on average winning most fights. As winners they took land and expanded.

This expansion created more strife, and expansion, leading to dominance of Italy. As a rising power, cities in southern Gaul (France), Spain and Sicily sought Rome’s assistance, and as a land power it allied itself with Carthage, a sea power. This arrangement came to an end when Rome supported Greek cities in Sicily against Carthaginian encroachment. A win in the First Punic (Carthaginian) War put the Romans on the path to empire, which was consilidated with the extirmination of Carthage in the Third Punic War in 146 BCE. At this stage Rome dominated the Western Mediterranean.

After their victory in the Second Punic War in 202 BCE Rome set out to punish Macedonia for aiding Carthage during that war. This dragged them into conflicts in the rest of Greece, which extended the empire there, leading further and further as other cities solicited their help in local disputes in the 2nd and 1st Centuries BCE, which swallowed up the eastern Mediterranean and Middle East.

Once they started getting dragged into troubles in the west, then in the east with the Hellenistic Kingdoms it gained a momentum which they had little control of, even if they wished, and ambitious generals were always anxious to go further. It was brought to a halt when Augustus, after the Civil Wars, put a stop to expansion and consolidated the empire within a defensive border of the Rhine-Danube and the Euphrates Rivers. This worked for a couple of centuries.

Addition:

The government of Rome developed into an empire by gaining territory mainly from wars, but at times from treaty and inheritance.