The statement is from The Information by James Gleick, claiming that there were no significant developments in logic between the Ancient Greeks and George Boole's The Mathematical Analysis of Logic in 1847.
Assuming this is true, what is the wider background to this breakthrough? Why did thinking start moving again after two and a half thousand years of stagnation?
This is obviously not true. Just a brief mention of the Scholastics, the humanists, the Renaissance, and all of the philosophers of the Enlightenment and the Scientific Revolution should be enough to prove that there were breakthroughs in that period.
Even the "Dark Ages" were not devoid of philosophy and reason. Ockham's razor, or the Law of Parsimony, is one of the most useful tools or heuristic devices in science and logic, stating that "pluralitas non est ponenda sine necessitate, 'plurality should not be posited without necessity.' The principle gives precedence to simplicity: of two competing theories, the simpler explanation of an entity is to be preferred. The principle is also expressed as “Entities are not to be multiplied beyond necessity.” Ockham's razor was crucial in the discovery of the heliocentric universe theory of Copernicus (which is also included in the period from Aristotle to the 19th century). William of Ockham lived from 1285-1349, the High Middle Ages, not usually a period associated with the development of "reason".
Any historian of philosophy would recognize that statement by Gleick as a huge overstatement, if not misinformed, probably driven by a disdain for any philosophy that is not sufficiently "analytical" (as in ignoring the Continental tradition). Even so, not recognizing the contributions of Isaac Newton or Galileo to the development of Reason...that's downright misleading.