What did Rome, "The City", do to become, "Rome", the empire?

I am going to assume you are referencing the identity transition from being a city with power over neighbours to an empire where anywhere within its borders were considered "Rome".

In simplest terms, the reason behind this was the Romans made the identity of being Roman exportable.

I think the best way to illustrate this difference is to compare Rome to its nearest historical predecessor, Athens. After the Persian war Athens set up a league of states with itself at the head with all states contributing funds to Athens for joint protection against any future Persian attack. In a similar example Rome leads an alliance of Latin cities set up to defend against the Gallic & Samnite incursions. Even though the Persian threat dwindles Athens keeps this alliance in place & uses force to ensure the members keep paying what is now clearly tribute to Athens. This change turned the alliance of defence into an Athenian Empire & the money was used to build the wonders of the Acropolis. Similarly, Rome incorporated Latium cities into its own state to become the preeminent regional power in central Italy.

However here is the difference, Athens hoarded it's citizenship & so remained a city with an empire of subject/tributary cities. Each city retains its identity & when Athens loses its controlling power there are no bonds holding this Empire together. Rome on the other hand had a much more foresighted approach to its allies. It required allies to provide 50% of troops but agreed that the allied troops would get a 50% share of moveable wealth when they took a city or defeated an enemy army. This is generous & makes allied soldiers enthusiastic in joining military expeditions with Rome. They allowed allied commanders high rank in the military & crucially they allowed the ruling class of allied cities to become Roman citizens. No where is this better demonstrated than by how the core central Italian cities stuck by Rome during Hannibal's invasion. They were invested in the Roman project & it's rewards were higher than independence for those in power in the cities.

Through organically created ad-hoc arrangements like this they created a ladder of relationships with its allies that became applied to it's Italian conquered cities. Now each city had its own individual deal but it roughly went something like this:

Tributary City: this was typically a newly conquered city that had to pay tribute to Rome & give 50% of its troops to the military as auxiliaries.
Allied city: a city who have agreed a military alliance with Rome, either immediately or after being conquered as a tributary for some years. Typically these would get Italian rights which gave ability to command & serve in normal Roman legions.
Latin Rights: this was basically everything the same as Roman citizens apart from the ability to vote in Roman elections or stand for them.
Roman citizenship, until the Social War this was restricted to Romans in the city of Rome or one of its colonies - colonies set up from Romans retained Roman Citizenship.

Through the decades, again organically as Rome face new threats & made better relations with its Italian neighbours the cities progressed through these types of relationships like a ladder. A once conquered city would start a tributary, then after a decade of tribute & no rebellions would become an ally. Loyal service as an ally would lead to grants of Latin rights. Due to the ladder system that naturally evolved Italian cities eventually expected to climb the last rung, Roman citizenship. Remember it's not just Rome with Roman citizenship but Roman colonies too which surrounded many Italian cities (Roman colonies were originally in strategic strong points to help dominate central Italy). Italians saw neighbours with more rights than themselves. This is initially denied by the Romans causing the "Social War" (war of the allies). The outcome is the enfranchisement as Roman Citizens for the Italian allies. This is important as it establishes firmly that Roman Citizenship is an aspect of the imperium & not fixed to a single city origin or blood. In some ways you could consider the outcome of the Social War the creation of the Roman Empire.

There are many other ways Roman Citizenship spreads, from the slave system (the lowliest of slaves in Rome could buy their freedom & their children would be Roman Citizens with more rights than the noblest of one of Rome's oldest allies), military land grants, merchants. With empire Rome would draw in new potential citizens and churn them out to colonies at settlers or new provincial elites. The stability of hundreds of years of peace allows this to create Roman citizens all over the Empire.

The next aspect is Rome's ability to create hybrid cultures that are predominantly Roman. That isn't to say "Romanization" occurs. That has long been debunked, provinces do not become homogeneous Romans. Rather the extra power & prestige of Roman society as well as the citizen rights outlined previously coops existing provincial elites into adapting aspects of Roman culture they like. They don't always adapt the same aspects in the same way, a 2nd century British town would bear little resemblance to a town in Syria, or even Roman Greece. Recent studies have also shown big differences in terms of culture between military installations & nearby provincial towns. The Roman military tended to have its own culture right down to its own gods. However the important thing to remember is all these new hybrid cultures were a form of "Roman" culture, with Roman the dominant identity in the hybrid.

TL;DR: It happens gradually through two main causes; The notion that Roman citizenship is a form of the imperial realm & not the city of Rome cemented in the end by the Social War; Secondly the ability of Rome to form hybrid cultures in its provinces with "Roman" as the dominating identity. The latter occurs largely because of the former as well as Roman organisational brilliance that makes adopting Roman ways more enticing.