The Roman Republic expanded because it was of benefit to both the middle-ranked military officers and those in the upper echelons of Roman political power.

Middle-rank officers of the Roman army were usually ambitious men trying to progress upwards in the Roman hierarchy. This was done by moving through the cursus honorum, a sequential ladder of military and political offices that ambitious young upper-class Roman men would try to hold. Most offices along the cursus honorum had minimum age restrictions, and it was both prestigious and practical to attain these offices in the first year you were of age. The sooner one completed a term in an office, the sooner one could advance to the next 'rung'. Progression could be made during peace time, but officers in the administration of a victorious conquering army would receive extra attention and prestige.

At the top of the primary line of the cursus honorum was the position of Consul. Only two Consuls were chosen per year, so the competition was extremely high. After a year as a consul, one would usually receive an appointment as a proconsul governor of a province. (I just want to provide a caveat here that my specialty is Roman politics during the imperial period. I am fairly certain everything I am saying here is true, but maybe someone with a specialty in Republican politics will be able to provide more detail.) A proconsul was essentially the king of their province. They no longer held the political power in Rome of a consul, but in their governed province they were the absolute authority. A proconsul had many motivations to wage a war of conquest: they commanded the armies stationed in the province and could raise new legions and auxilia at will, as long as they were able to pay for them. And there was no better way to pay for new armies than by using them to conquer new lands for Rome! A proconsul could also use wealth gained during conquest for personal matters. Julius Caesar used his time as proconsul of Gaul to repay much of his debt to Crassus, for example. Furthermore, a significant victory over a foreign enemy would usually mean the proconsul would be awarded a Triumph, one of the most prestigious awards a Roman could receive. A Triumph was a massive ceremony in Rome where a proconsul's army would march through the city, displaying conquered enemy leaders and seized booty. Finally, a proconsul's standard term was one year, but this could be extended if the proconsul was commanding a campaign. Thus, a proconsul had many motivations to wage war.

As you can see, those in control of the Roman military had a great deal to gain from wars of expansion. A desire to advance up the cursus honorum was ingrained in young Roman men; a huge part of Roman culture was a need to outdo both your ancestors and peers in terms of power and influence. This culture, combined with the massive financial rewards that could be gained through a successful proconsular campaign, leaves no surprise that the Romans conquered just about everything they could. As my answer here details, Roman expansion in Europe only really stopped when it was no longer financially beneficial.

In the very early days of the Roman Republic, it was just one minor city state in a world of minor city states. It fought against its neighbours, because that is what minor city states did. It won some battles, lost others. It seemed destined to be just another footnote in the histories of the period.

The moment of change, according to the Roman historians such as Livy, was when Rome managed to conquer the neighbouring city of Veii in 396 B.C., after a ten year siege.

... yeah, Livy was probably just trying to evoke the Trojan War here. The whole event was thoroughly mythologised by the Romans.

The significant bit is this: where other city states had made conquered states submit and pay tribute, Rome annexed Veii and created new tribes of Roman citizens from the territory of its former foe. Rome did not just gain victory, it grew.

Why? How did they manage to to this? We just do not know. We have no good written record of this period, and the histories we do have were written much later, and at best we have Roman historians who read earlier Roman historians that are now lost to us. There are a few bits and pieces of inscription, but nothing that can really give a satisfying explanation.

But suffice it to say that Rome did start to expand early, and kept it up. It gained a dominant position in the region of Latinum, and as it went through the usual pattern of victories and defeats, it continued to somehow incorporate former enemies into its imperial project. More distant cities were not annexed or made to pay tribute, but were instead turned into "allies" who could manage their own internal affairs but had to follow Rome's lead in foreign policy, and supply troops to Rome's army. By the mid Republic, fully half or Rome's armies were made up of these Italian allies, with the other half being the Roman legions themselves. Although there was some friction between Rome and its "allies", and later on even one full-blown war that was the greatest threat to Rome's survival since Hannibal, on the whole this system worked extremely well. The Italians partook in the glory and got some of the gains of Rome's conquests, without losing too much of their autonomy and dignity.

As Rome became more powerful in the region, it also started to be noticed by greater powers. A conflict with a Greek colony in southern Italy lead to intervention and invasion by the Epirote king Pyrrhus. Conflict over a city in Sicily (among other things, this was just the trigger) led to conflict with Carthage. Greater power led to greater enemies, and these no-longer-regional wars led to Rome first gaining provinces outside of Italy.

After Rome had become the most powerful state around, the factors you describe started to play a role: Roman noblemen's desires for glory and power through conquest. By the 2nd/1st century B.C. Rome was by far the most powerful state in its part of the world, and conquest largely became a matter of motivation rather than ability.

But these big wars and big conquests were only possible because of all the little gains made in the preceding centuries, where Rome had managed to somehow get most of Italy on board. It was thanks to this process that Rome had the kind of manpower and resources that allowed it to survive disaster after disaster at the hands of Hannibal, and still come out on top.