The basic answer here is that the Black Death caused such a huge upheaval in medieval society that the Renaissance was able to emerge from the chaos.
The Bubonic Plague, of course, killed huge numbers of people. This led to such things as shortages of labor. When labor was in short supply, the laborers (who were, of course, from the lower classes) were able to demand better wages and more favorable terms for selling their work. This helped to overthrow the old system in which the working classes were completely subjugated.
In addition, the Black Death led to the breakdown of old ways of thinking. This is only natural given the circumstances. When such a huge proportion of the population dies and no one can understand it or do anything about it, faith in the old ways will naturally diminish.
By undermining the old social structures and the old ways of thinking, the Bubonic Plague helped to lead to the Renaissance.