Constantinople fell in 1453. The Ottoman Turks took it over. When this happened, many intellectuals moved from Constantinople to Europe, such as cities like Florence. When they did this, they brought their learning and expertise, which they honed since the days of Constantine. These intellectuals helped the Renaissance in a few important ways.
First, Greek intellectuals brought a new way of thinking that was apart from the scholasticism of medieval Europe. This allowed people to see outside the type of thinking that was dominated by the dogmatic doctrines of the church.
Second, Greek intellectuals brought many manuscripts from the ancient world. In a time without a printing press, this was enormous. This allowed Europe to go back to the glories of the classical world. Moreover, as the Renaissance accelerated, a new boldness developed that would even challenge the glories of the ancient Greeks and Romans.
Finally, the fall of Constantinople brought a passion for philology. This caused Europe to rediscover the ancient world in greater detail. For example, the Protestant Reformation would not have taken place unless there was a new emphasis on the study of original languages. For the Reformation it was the study of Greek and Hebrew.
All of these points greatly accelerated what was happening in Europe. However, I do not want to give the impression that the Renaissance was only owing to exiled Greek intellectuals. Greek intellectuals did help, but a new movement was already brewing for a hundred years.