One of the best and earliest extant comments about Cicero's eloquence is found in Quintilian's Institutiones Oratoriae. Therein, he delivers a defense of the claim that Cicero bests any other Latin orator (and only Caesar gave him a run for his money) and even rivals Demosthenes for the title of greatest orator of all time. The passage is long, but I'll remark on some of the key points.
Of their great excellences, I consider that most are similar—their method, their order of partition, their manner of preparing the minds of their audience, their mode of proof, and, in a word, everything that depends on invention.
So much of Cicero's fame rests on the substance of his speeches and how they arranged them to be persuasive.
In their style of speaking, there is some difference. Demosthenes is more compact, Cicero more verbose; Demosthenes argues more closely, Cicero with a wider sweep; Demosthenes always attacks with a sharp-pointed weapon, Cicero often with a weapon both sharp and weighty; from Demosthenes nothing can be taken away, to Cicero nothing can be added; in the one there is more study, in the other more nature.
It's not that Cicero's style is the only path forward; Demosthenes and Cicero do differ, after all. But where they differ, they each excel in those areas. Cicero can wax eloquent. His verbosity is like a wave crashing down. There is little room for confusion or ambiguity. (Contrast that with someone like Tacitus, who's often too terse.) This is what Quintilian means with "weighty" - his later speeches might not be pithy (though you can find that in some of his earlier ones), but he leaves little room for doubt.
In wit, certainly, and pathos, two stimulants of the mind which have great influence in oratory, we have the advantage. Perhaps the custom of his country did not allow Demosthenes pathetic perorations. But on the other hand, the different genius of the Latin tongue did not grant to us those beauties which the Attics so much admire.
This is one of the major stylistic camps in ancient Rome: Asianic or Atticist. Cicero actually navigated a good middle ground. As I wrote before:
Cicero’s style was a blend of both the Atticist and Asianist school. His earlier orations were full of ornament and short, pithy figures. Over time, though, he slowly acquired more Atticist tendencies, with some of the periods in, e.g. the Res Publica being quite long with several layers of hypotaxis. Over the course of his career, he refined the construction of his works in favor of symmetry and elegance, placing him high above almost everyone else in eloquence.
This "best of both worlds" approach escapes the pitfalls of each camp: too much rhetorical flourish can feel vapid, but too structured can feel dry.
Nor did he, by zealous effort, attain only what was excellent in each of these, but drew most or rather all excellences from himself, by the felicitous exuberance of his immortal genius.
Quintilian means that Cicero is highly original and innovative, that he isn't just copying the old orators of yore, like some must have been doing.
In all that he says, indeed, there is so much authority that we are ashamed to dissent from him. He does not bring to a cause the mere zeal of an advocate, but the support of a witness or a judge... It was not without justice, therefore, that he was said by his contemporaries to reign supreme in the courts, and he has gained such esteem among his posterity that Cicero is now less the name of a man than that of eloquence itself.
It should be remembered that Cicero was the Roman equivalent to a lawyer, and that many of his speeches were suited for a trial. Ultimately, what matters most is the persuasiveness. He was so persuasive that Hortensius, his senior and quasi-mentor, quit trial law after he lost to Cicero in Verres' trial.
Cicero was also well known for his portraiture. As I wrote before:
He was most successful in creating a realistic portrait of both those he defended and attacked, using every poetical device in his oratory to make the jury actually feel as if they knew the person well. It was so effective that after only one day of presenting evidence in his corruption case, Gaius Verres, a favorite among the elite, immediately went into self-imposed exile, sure of Cicero’s success.
Not everyone loved Cicero, of course; he had his fair share of detractors. But the weight of Quintilian (and even the deference that Seneca, whose own style was more fashionable in the middle of the 1st century CE) helped establish Cicero was the name for eloquence, or as Quintilian put it:
Cicero is now less the name of a man than that of eloquence itself.