I'm afraid any answer to this question must begin by considering what is understood to be the 'Renaissance' and the 'Scientific Revolution'. And that consideration, in turn, inevitably reveals a number of historiographical difficulties.
The first of these is that neither of these were 'events', at least, not in the sense of a war or an assassination. They have been used to signify shifts in intellectual pre-occupation. In the case of the Renaissance (which arguably occurred from the fourteenth to the early seventeenth centuries, and was NOT exclusive to Italy as some on this forum have suggested), scholars were increasingly concerned to formulate an authoritative corpus of works from classical authors, and to produce a more sophisticated appreciation for Greek and Roman customs and language. In the case of the Scientific Revolution (which is generally associated with the late sixteenth and seventeenth centuries) the drive was rather different. Rather than seeking to recover or reconstruct knowledge from the past, intellectuals turned their attention to the formulation of NEW, observable, and experimentally driven analyses of the natural world. It was, on this telling, the transition from Renaissance to Scientific Revolution (and eventually to Englightenment) that proved so foundational to our present, modern world.
Both of the claims I have described above are no longer uncritically accepted by any historian. They continue to be used as conceptual placeholders, but the Renaissance was not a straightforward reconstitution of ancient knowledge, and the scientific revolution was neither revolutionary nor was it scientific.
With regards to what could be called the Renaissance, there certainly was a drive to get away from the prior reliance on Latin translations of Arabic translations of original texts, and the integration of texts that were demonstrably the work of scholars in the 10 century ce (like Pseudo-Aristotle) rather than 5th century bc. But this process was not one devoid of seemingly 'scientific' endeavors, a point to which I will return in a moment. The central drive to do this sort of work stemmed ultimately from Catholic and later Protestant anxieties over intellectual authority. The drive from within the Church itself, and then the influence of wealthy patrons across Europe, to formulate a definitive and orthodox understanding of the past was ultimately the mechanism by which scholars eventually (after more than two centuries of tearing out their hair) were forced to conclude that they could not in fact write history in any definitive sense, try though they might.
The point about evaluating and understanding ancient authorities through seemingly scientific experiments is something that is very familiar to academic historians of science. This is why the Renaissance and so-called Scientific Revolution are in fact two terms for the same process—of generating new information and new knowledge in order to recover ancient wisdom. Or, at the very least, to sort out the wheat of the ancients from the chaff of modern innovation.
The 'real' historical picture of this development is not at all clear for a very simple reason. To adequately understand how intellectual change in any form occurred over time, historians should be looking at each individual scholar—their particular institutional and political circumstances, their own private musings about what they were up to, and the way they justified their enterprise publicly. No historical category—certainly not the Renaissance or Scientific Revolution—survives a rigorous consideration of the past at such a granular level. But that's why asking how the Renaissance gave way to the Scientific Revolution is approaching an important historical process from the wrong standpoint.
The point about this dynamic of increasing intellectual activity, is that people were desperately trying to bring together a range of discourses that we now see as different, if not totally antithetical. And our perception is the inheritance of their failure to make science, history, and indeed the humanities, work for religious ends. That is the legacy—and we would be well advised to understand it on its own terms before we start trying to do the same thing while expecting a different result.
