Although this is also an art movement (see Neoclassicism), it started as a literary movement. Among intellectuals during the late 14th and early 15th centuries, there was a desire to return to the values taught by the philosophers of classical Greece and Rome. Aristotle, Sophocles, Plato... they all became in vogue again. This was partially a response to the turbulence of the 14th century, which included the Hundred Years War, the conflict between England and Scotland brought on by the death of the child-maiden who was to be queen of Scotland, the internal strife in Italy leading to domination by the Medicis, and, of course, the Black Death. All of these social factors combined to create uncertainty in modern (i.e. late 14th and early 15th centuries) institutions; the prevailing thought seemed to be, "Our government and religion have failed us; let's turn to the ways of the ancients for an answer."
This was, of course, antithetical. The ancient philosophers that they read didn't have any real answers to problems from their own time periods, and they certainly didn't have answers to the problems facing the late 14th and early 15th centuries. This didn't stop these modern philosophers from elevating the ancients to god-like status; they even started differing schools of thought about the merits of original Latin versus modern, Church Latin.
The movement offered no real solutions, but it did influence several important figures of the time. Petrarch, one of the early neo-classicists, wrote of the world he lived in because he believed that an appreciation of art was a key element of the classical civilizations. The Florentine scholar Bruni wished to found a government based upon the Roman ideal of representative government (most neo-classicists chose to ignore the fact that Rome had become a dictatorship despite its so-called "perfect" government). The Dutch writer Erasmus was one of the later neo-classicists, and his philosophical writings, many of which influenced Martin Luther, were all derived from classical thought. It also is the reason why today we are able to read the works of the Greco-Roman masters; without the Renaissance translations, many of the texts would not have survived.