Rome's long slide into collapse was, as Gibbon contended in this quote, fueled by its unprecedented success expanding its borders over most of Europe, North Africa, and the Middle East. We can see this in a number of ways.

1) Imperial Overstretch: The larger empire became, the longer the borders it had to defend and the more numerous and powerful its enemies became. The border with Germania along the Rhine became a continual zone of conflict, for example. The Battle of the Teutoburg Forest in 9 AD, when three Roman legions were ambushed and slaughtered, was a major flashpoint in this conflict. Though the Romans responded with punitive expeditions, Germania was never subdued and became a wound that never healed.

A century after Teutoburg, the Emperor Hadrian became the first Emperor to recognize that Rome could not expand indefinitely. He abandoned Trajan's conquests in the Middle East and pulled back the border of Dacia. His foresight probably bought Rome centuries. Choosing and fortifying the Empire's borders rather than having them set by military misadventure stabilized Rome, but it also began the process of military decline as the legions were filled out by troops who were less experienced in open battle, less physically fit, and less connected with the home city than their predecessors had been.

Later, the Danube frontier became the weak point. The Goths wiped out a Roman army in 378 and killed the Emperor Valens. It was a defeat the Empire could not answer, and Gothic settlement became a fait accompli. The Goths helped fortify Rome's borders but did so only when it suited their interests. In 410, Alaric, a Gothic chieftain, sacked Rome itself, and in 476, Odoacer deposed the last Western Emperor.

2) Cultural Irrelevance: At first, it was a matter of the distance. Roman emperors stopped being chosen from the leading citizens of the city and instead were foreign generals, mainly from Illyria. These hard men had no love for Rome itself. Rome looked down on them as provincials—half-savage Barbarians who lacked Italian education and refinement.

One Illyrian, Diocletian, became the first to split the Empire in the late third century, first into two pieces, then four. His idea was to always have an emperor on the scene to meet threats. It worked while he was alive, but the system disintegrated into a series of civil wars.

The winner of these wars, Constantine, saw the truth: the empire's future lay in the east. The east was wealthier, more populous, and closer to important trade routes. Additionally, Christianity was spreading throughout the empire as the common people of Rome abandoned their Pagan faith. Lack of social mobility and economic prospects brought on a hopelessness that proved fertile soil for a religious preaching salvation and the equality of all before God.

In 330, Constantine moved the capital from Rome to Byzantium, and Rome never recovered. When the Empire was finally divided for good in the late fourth century, Rome became the capital of a Western Empire that was left to rot by the wealthy, Christian east.

Soon, Rome was too far away to even fend off invasions from Gaul and Germania (let alone Persia and Dacia), so the western emperor moved his seat to Ravenna. Lacking the manpower and money to defend its borders, the Romans turned to foreign armies like the Goths to fight for them. Then, lacking the strength to even administer its domains, Rome abandoned them, leaving the Franks to administer Gaul and the Britons to administer Britannia. Rome, as imperial capital, effectively died poor and irrelevant.

3) Economic Rot: Conquering one eastern kingdom after another led to an influx of gold, plunder, and slaves to which the Roman economy became addicted. Thousands and finally millions of slaves greatly exacerbated inequality. The wealthy who had them no longer needed to employ free men to work for them. Prosperous farmers bought up the lands of their neighbors and claimed all public lands for themselves. This led to an influx of poor, unemployed men and women into the city itself.

Inequality in turn became the source of the instability that brought down the Republic during the first century BC. As Roman patricians became wealthier, they became more resistant to reforms that became ever more necessary. Reform-minded politicians resorted to demagoguery and unconstitutional maneuvering to try to overcome the Optimates. This led to an escalating cycle of political violence. First the Gracchi were killed, then Marius and Sulla became the first generals to march on Rome, massacring each other's followers.

4). Military Corruption: The legions of the Roman republic that defeated Hannibal in 202 BC were made up of citizen-soldiers who had to pass a property qualification to serve. 150 years later, there were too few Romans who met the property requirement, and those who did were too wealthy and self-interested to serve. The qualification was waived and the legions became a source of employment for the poor. The soldiers of the late republic no longer served out of patriotic duty but for economic survival. Further, these soldiers acquired plunder from the victories of their commanders. As such, a successful general commanded more loyalty than did the republic itself. Thus, Marius and Sulla marched on Rome with armies loyal to them, personally. Later Caesar, Crassus, and Pompey combined their wealth, influence, and military might to overcome Optimate resistance in the Senate to reforms.

While the First Triumvirate lived, an uneasy peace and stability was maintained, but with the death of Crassus, the final civil wars of the republic soon followed until Julius Caesar triumphed over Pompey and the Optimates. After Caesar's assassination, his heir, Augustus, ended republican governance for good and became the first Roman emperor. His wisdom and strength allowed Rome to prosper during his lifetime, but the inherent instability of monarchy began to show itself in his successors, particularly Tiberius, Caligula, and Nero. Rome was strong enough to survive these kinds of rulers and was rewarded by the century of peace and prosperity that Gibbon referred to as the "Five Good Emperors." When Commodus succeed Marcus Aurelius, however, a period of endemic political instability set in that began the inevitable process of decline.

Essentially, Gibbons is arguing that the decline and fall of the Roman empire was a consequence of its corruption from within. This in turn was a consequence of a number of factors. First, he writes that the Empire had become so large that its leaders and its citizens lost the sense of public zeal and patriotism that had made the empire so powerful and stable in the first place. This view was common among many thinkers of the Renaissance and the Enlightenment. Modern historians associated it with the concept of "republicanism," which emphasized the tenuous nature of societies that depended on what was often understood as "virtue." In addition to wealth, Gibbons argued that Romans lost their virtue due to geographic expansion and (most controversially in the eighteenth century) the advent of Christianity. This new religion eclipsed the old cult of the emperor, and its focus on spiritual rather than secular matters was anathema to Roman virtue. The weakened Empire was then, over time, torn apart by invasions by "barbarians" from outside its borders. But this quote, focusing as it does on "immoderate greatness," points at only one aspect of Roman decline. It argues that the Romans, having become wealthy, became soft and corrupt, turning their backs on what made them so powerful in the first place. Thus, the empire "yielded to the pressure of its own weight" before it was finished off by external forces.

Here, Gibbon is making the claim that the elements that made Rome a great power were eventually the ones that led to its eventual and inevitable downfall. He is placing the blame for the fall of this empire not on outside forces, such as barbarian invasions, but on Roman hubris itself.

Throughout its many centuries of growth and prosperity, Rome built something that was not sustainable. Its borders were so far-flung that they were impossible to defend. Its population was so large that it was impossible to feed. Its lifestyle was so lavish (at least for the upper classes) that it became impossible to afford.

Gibbon is making the argument that no empire, not even the mighty empire of Rome, could grow and prosper forever. There comes a point when something becomes too large to succeed. By the time that Rome collapsed, it could no longer raise the armies it needed to defend itself. Its administrative apparatus was stretched to the breaking point as effective communication across such a large dominion was near impossible.

Gibbon wrote his work in the late-eighteenth century. At the time, Great Britain had built a mighty empire of its own. He wanted his work to be a warning and a check on British imperial ambition. He, and many of his contemporaries, saw parallels between Great Britain and Rome. Here, Gibbon provides a warning that even the mightiest of empires can become too large and successful to sustain themselves in the long run.

Gibbon is making the argument that the elements which allowed for the greatness of Rome could not be the forces which sustained it.  The implication is that the rise to power and the judicious exercise of it are not one in the same.  This allows for a stronger understanding of why the Roman Empire was, after a point, unable to be sustained.  The natural state of expansion caused it to be untenable and nearly impossible to defend.  This would be an example of collapsing under its own weight.  At the same time, questionable decisions on the part of its leaders differentiated it from the sound and wise leadership of those who allowed Rome to be great.  At the same time, the belief in its own greatness, something that impacted its ability to be reflective enough to embrace change, is what prevented the empire from modernizing and advancing.

Basically, what he's saying here is that the Roman Empire was a victim of its own success.  He's saying that it got too big to be able to survive.

So when he says "immoderate greatness" that means that it was overly great, not moderate in how big it got.

When he says that the "causes of destruction multiplied with the extent of conquest" he's saying that as Rome conquered more and more land the causes of its destruction were exacerbated.

When he says that it "yielded to the pressure of its own weight he is, again, saying that it got too big and too ponderous.

To Gibbon, a major cause of the fall of the Roman Empire was that it got so big that it cost too much to run and it cost too much to defend.  That's what he's saying in this quote.