Why or how was the return of Platonism in the Renaissance at the expense of Aristotle?
While it is largely the case that Neo-Platonism flourished at the expense of Aristotelianism during the Renaissance, it is important to realize that much of Aristotle's decline is largely a matter of self-identification among Renaissance philosophers. Disdainful of the esoteric theological questions that obsessed the Scholastics, who were admirers of Aristotle, these new thinkers embraced neo-Platonism, which seemed to address more expansive questions about the world. Petrarch, in particular, styled himself as an anti-Aristotelian and a neo-Platonist, though he also quoted from the Nicomachean Ethics.
Yet they (including Petrarch, a proto-Renaissance thinker) also accepted much about Aristotelian epistemology (the study of knowledge). They accepted, for example, that human knowledge is not completely prior to existence, a hallmark of Aristotle's thought.
But two of the major elements of Renaissance thought were self-consciously Platonic in nature. The first was the idea of a Great Chain of Being descending from divine perfection. Another major aspect of Platonic thinking that re-entered Western thought in the Renaissance was the notion of ideal forms, a doctrine best expressed in Renaissance art and architecture, which emphasized ratios and balance with an almost mystical zeal. Hermetism, a movement that was decidedly neo-Platonic in nature, counted Leonardo da Vinci among its practitioners.
But ultimately, Renaissance thinkers such as Pico Della Mirandola and others in many ways plundered the ancients for whatever they could find, even if they tended to identify with one thinker or another. So Aristotle was far from forgotten, though his stock dropped among the mainstream of Renaissance thought.