I assume that the specific quote you're thinking of is Gibbon's statement that "If a man were called to fix the period in the history of the world, during which the condition of the human race was most happy and prosperous, he would, without hesitation, name that which elapsed from the death of Domitian to the accession of Commodus" (DF I.3.II). It's worth considering the specific way in which Gibbon phrases this statement; if "a man were called to fix" is not the same as Gibbon passing his own opinion. Broadly speaking, Gibbon acknowledges the positive features of this late imperial Rome; its stability, wealth, devotion to legal progress - but in this regard I'd argue that Gibbon is not expressing his own view; for, in accord with Tacitus, his favoured historian, the Roman Empire suffered for the fact that its citizens lacked personal freedom; for a state to be more successful than that Rome, it would have to combine the aforementioned features with a certain level of popular involvement, of constitutional guarantee. Not coincidentally, this applied to England (with its parliament, in contrast to monarchical rule in France and Spain); Gibbon was far more an advocate of eighteenth century England as the best era in human existence - an unsurprisingly Enlightenment view.

Second century Rome was ushered in by the deaths of Domitian and Nerva. The former established the pillars for a peaceful Rome by becoming Caesar after the bloody battles during the "Year of Four Emperors." His policies were considered ruthless internally, but he was dissuaded from warring further due to the tumult of the times. Domition was assassinated after a rather lengthy 15 year rule.

Nerva became emperor in 96 AD after the Senate unanimously appointed him. From here began a prosperous 100 year period spanning the entirety of the 2nd century, known as the Nerva-Antonine dynasty. Although Nerva's reign was short at just 2 years, he made peace with the different parts of the army and brought unity to the empire. After he died of natural causes, his son Trajan took the throne. Trajan is considered one of the greatest Roman emperors because he presided over Rome's territorial height. Other than a 4 year long war with Parthia (whose campaign ended with Trajan's death), times were relatively peaceful for the Romans. The Pax Romana marked this era until the end of the 2nd century.

Trajan's time as a civil emperor is what made him revered among the Roman public. Rome was wealthy at this time following the Roman conquest of Dacia. There was a lot of loot to be had from these wars, namely in the form of gold from sacked gold mines. This wealth allowed Trajan to construct a number of civil improvements, such as his eponymous forum and market. He also built many triumphal arches around Rome, and he reconstructed a lot of key roads. In 107 AD, he hosted a 4 month long gladiatorial festival which attracted millions of spectators from all over the empire.

The next 80 years were culturally rich for Rome. Its emperors were scholarly, and valued knowledge and wisdom throughout their empire. Instead of engaging in expansionary wars, they promoted improvements from within the empire. Hadrian exemplified this with his consolidation of the empire stemming from his concession of Mesopotamia and Armenia.

It was a great time to be alive as a Roman. You had the security of being ruled by multiple good emperors in succession. The supposed tyrants like Nero and Caligula were long dead. Poverty was reduced (among the middle class) as the treasury was overflowing with gold. There were no worries of being purged or having your home and money taken away to line the emperor's pockets. You were living in a time where the future looked promising and the past was archived. Some of the greatest writers and historians were alive, such as Tacitus, Suetonius, and Plutarch. Hellenic culture was spread throughout Rome during the rule of philhellene Hadrian. It was a great time to be alive as a free Roman citizen with some economic standing.

 

Here are my sources which I recommend you read if you are interested in the history of Ancient Rome.

 

Simon Baker - Ancient Rome: The Rise and Fall of an Empire. An excellent companion to the History Channel's documentary on the subject.

Edward Gibbon - The History of the Decline and Fall of the Roman Empire. I'm sure you are familiar with this text as you referenced Gibbon. Chapter 1 from this text deals with Rome's extent throughout the 2nd century AD. Chapter 2 talks about the reasons why Rome was prosperous during this same period.