There is quite a big difference between how the Romans justified their expansionism, and the reality behind their propaganda. If by “justify” you mean “what were their reason”, then the answer can be pretty much summed up, with a few exceptions, into three categories: Money, Resources and Glory. First, however, in terms of actual justification, it is hard to get past the assertion that the Romans were simply spreading the Pax Romana and their superior culture to less fortunate “barbarian” nations.

Their actual motivation for their expansionism was, as you might suspect, much more complex than a simple desire to better the world and liberate the wretched barbarian from their self-imposed ignorance. In the early days of the Roman state, when the city was still just one of many disparate city states that were dotted across Italy, expansion was predominantly reactionary. It was, as you suggest in your question, a desire for security, (which came about due to the frequent minor wars being fought by Rome with its neighbours) that ultimately led to the absorption of external aggressors such as the Samnites as Rome sought to secure its tentative position.

Later in Roman history, and particularly during the days of the Republic, expansion was usually motivated by the three factors outlined above: money, resources and the desire for personal glory/power (imperium). Both Julius Caesar, who conquered Gaul and parts of Hispania, and Gaius Pompey, who brought Asia Minor and the Levante under Roman control, were motivated both by the desire for personal glory, as well by the prospect of enhancing their public and political image, and the opportunity to make vast amounts of money to fund their ambitions – it is well documented how close to bankruptcy Caesar was before his Gaulish campaigns. In contrast to what you might expect regarding Roman expansionism, Caesar’s conquest of Gaul was considered completely unnecessary and even illegal by many members of the Senate in Rome and calls were even made for him to be arrested and tried for war crimes – although you can be sure this was out of political opportunism on the part of his enemies rather than a desire to avenge the million plus Gauls who died directly as a result of Caesar’s actions. Despite political opposition, Caesar continued to conquer Gaul and even attempted to do the same in distant Britain (little more than a myth to the average Roman) predominantly because of the vast wealth (gold and slaves) it afforded him, and the public image it created of Caesar as a conquering hero championing Rome’s hegemony. Conquest also allowed Caesar, and others, to create massive standing armies of men resolutely loyal to their commander and paymaster rather than to the Roman state, something which was important if, as many historians suspect, his ultimate goal was always to gain ultimate power in Rome.

Thus we can see that conquest, in the days of the Republic, was predominantly driven by the individual rather than by the state, although the Senate sanctioned the majority of these endeavours, be it willingly or through political chicanery. There was often much bitter contest and jealousy between generals competing for these commands with their rivals, Marius and Sulla would be one example, Pompey, Lucullus and Crassus would be another. Lucullus even had his command of Rome’s campaigns against Mithridates in Asia Minor stripped and awarded to Pompey instead by the Senate. The desire of political and military figures to gain personal glory and wealth, as well as the magnificent honour of a Roman triumph, cannot be underplayed as a factor in Roman expansion during the Republic, although it was rarely if ever cited as an actual motivation. Usually some pretext was found to justify an invasion, for example Caesar cited the migration of a Germanic tribe called the Helvetti and the “threat” they posed to Rome as justification for his campaigns in Gaul, similarly Rome used a disputed will and the rebellion of Mithridates as an excuse to conquer much of modern Turkey and Syria.

On a side-note, the Roman obsession with the military triumph is well documented and the lengths that some generals went to be awarded them speak to the power that spectacle had in the Roman world. A triumph, awarded to a Roman commander who excelled in the field and added new territory to the Republic, was a great opportunity to showcase wealth and power and would ultimately benefit an aspiring general or politician greatly in his future career. The influence that conquest and a triumph could have on the Roman public is well documented, in the days of the Empire, the Emperor Tiberius was reportedly reluctant to allow his nephew Germanicus to extend the borders of the Empire, for fear of the popularity and image boost a triumphal conquest would afford his more popular relative (see Tactius’ Annals for this). Thus, we can see that conquest played a vital role in the development of an aspiring leader - consul or dictator in the Republic, Emperor during the days of the Empire. It is, consequently, clear why commanders frequently sought any excuse to raise an army of loyal men and expand the empire’s borders.

The desire for personal and public glory was undoubtedly a major factor in Rome’s aggressive expansionist policy, but it was not the only one. Some cases were slightly different, although motivations frequently overlapped. Pompey and Caesar, and later Anthony and Octavian oversaw the annexation of Egypt, a process which was predominantly a political rather than a military achievement (although there was still military involvement). The Nile delta was famously fertile and as the Empire and its population grew, Egypt’s grain production was vital to the support of the booming population back in Rome, many of whom relied heavily on a state issued grain dole, a subsidy which the previous “bread basket of Rome”, Sicily, could no longer cater for alone. Thus we see an example of conquest being out of necessity rather than opportunism or reactionism. Egypt continued to supply Rome with grain for nearly all of its history and it a mark of how vital it was to the stability of the Empire that Senatorial command in Egypt was heavily regulated by the Emperors, to prevent an ambitious senator seizing Rome’s grain supply.

Going back, briefly, to the idea of image being a major factor in conquest, after the advent of the Empire, conquest was often a means by which an Emperor could augment or improve his popularity and image with the Roman citizenry. Claudius, for example, was keen to dispel the public perception of him as a feeble idiot and thus embarked on Rome’s second, and this time ultimately successful, invasion of Britain. He then held a very public and spectacular triumph in Rome, despite having minimal involvement in the actual military invasion. Although the technical justification of the invasion of Britain was to spread the Pax Romana and to combat the alleged influence of British druids in Gaul, in reality the conquest was more of a PR stunt to boost Claudius’ shaky image, similar to how an unpopular politician today might be seen helping out at a soup kitchen or hugging babies at the mall: conquests and triumphs appealed to the average Roman’s sensibilities.

In contrast, the conquest of Dacia by Trajan between 101 and 106 AD was, I believe, more of a reactionary measure because of Dacian incursions into the Empire and much of the territory gained was abandoned again not long after, although I am not as familiar with this period of Roman history so I cannot swear to this. If true, it does however signal that at least some Roman expansion was motivated by a desire to protect the Empire’s borders, as you suggest in your question (I would also recommend looking at Roman interactions with the Parthians in relation to this). To summarise, at least in the context of the Republic and early Empire, conquest was predominantly driven by individuals seeking personal wealth, power and glory in order to support their political desires, although as with Egypt, the need for material resources also played a massive role in Roman conquests. The Romans often justified these practical conquests as them delivering civilisation and prosperity to barbarian lands but this was rarely the driving motivation behind Rome’s aggressive expansionism.

TL:DR Rome’s conquests were driven by a desire for wealth, resources and prestige. Although border security played a role as well.

-btw I’m typing this on my phone on my break at work so I can’t look up sources easily, but if you are interested and I managed to answer some of your question without going off on a massive tangent, I can find specific quotes to support my claims.