This is an interesting question, and there is a fair amount of scholarship on renaissance rhetoric (and its effect on enlightenment rhetoric after). But the two questions you've asked actually cut to the heart of things quite nicely. Depending on what you're after, I can recommend a few texts from which my answer is drawn:

A nice summary of the historical development: Heinrich Plett, "Rhetoric and Humanism," Oxford Handbook of Rhetorical Studies, ed. Michael J. MacDonald (Oxford, 2014/7).

A more wide ranging study: Brian Vickers, In Defense of Rhetoric (Oxford, 1995)

And since you have an interest in history: Nancy Struever, The Language of History in the Renaissance (Princeton, 1970).

Roman rhetoricians, rather than the Greeks, were the primary inspiration for the revival of rhetoric in the renaissance. Although Aristotle's Rhetoric was translated, most of the major innovations took Cicero's and Quintilian's works as their point of departure. Although the Romans were influenced by Aristotle (and Plato, to some degree), their closest Athenian progenator was Isocrates. Isocrates was much more sympathetic to the arguments of the sophists than someone like Plato (cf. the dialogue Gorgias for the most extended critique), particularly because he saw rhetoric as a critical component of proper education. But he distinguished himself from the sophists in several ways as well, most importantly by downplaying the scope of rhetorical education. Rhetoric for him was a method of addressing problems or conflicts through language, not only in politics or law, but even in the course of everyday life. But rhetoric alone was insufficient for a proper education, and required supplementing by study in other disciplines. Cicero and Quintilian followed the Isocratean tradition by suturing rhetoric to education. (Although Cicero's De Oratore is structured as a dialogue in the Platonic style, it is as much a defense of rhetoric from imagined Platonic critics as it is a pedagogical tool.)

The development of rhetoric in the renaissance took much from both Cicero and Quintilian, and the author of the Rhetorica Ad Herennium, who is likely either Cicero or Cornificius. Whereas Aristotle, I would argue, shows us the significance and form of rhetorical action in a civic context, the Roman texts provide a normative structure for rhetoric, rhetorical acts, and rhetorical education. Concepts from Aristotle like ethos, pathos, and logos were still important, but they took second seat to five "canons" of Roman rhetoric--inventio (invention), dispositio (arrangement), memoria (memory), pronuntiatio (delivery), and elocutio (style).

The revival of Roman rhetoric in the renaissance took form under the guise of humanism, which was itself a reaction to scholasticism. To be as reductive as possible, if scholasticism focused on imparting authoritative knowledge to students, humanism conceived of education as a means of human development. Hence the continued influence of Isocrates. Rhetoric was a crucial part of this, uh, renaissance because it was aimed at helping students both understand the speech/text of others but also produce it themselves.

The two renaissance figures most commonly associated with the development of rhetoric are Erasmus and Ramus. Erasmus's Copia was a popular text that offered extensive discussions of style, both in the context of elocutio and inventio. Copia focuses on the abundance of style (hence the title), which is a clear and direct response to the limitations of scholastic education (which was more dogmatic).

Ramus, on the other hand, is essentially responsible for the public image of rhetoric today as ornamentation. He divided the five Roman canons between rhetoric (elocutio and pronuntiatio) and dialectic/logic (inventio and dispositio), arguing further that memoria wasn't really part of rhetoric at all. The effect of this division is palpable. To be sure, it allowed scholars to focus on style on its own, which led directly to the development of literary studies. At the same time, by disarticulating style/delivery from invention/arrangement, it allowed style to be deprecated by emerging rationalists like Descartes.