The Medici were more than just a family, they were a core of influence in the renaissance. This influence initially came through the wealth of Giovanni di Bicci de Medici in his role as banker to the Pope. In those times the papacy was corrupted by its power and was hugely wealthy. It is hardly surprising that the bank looking after such wealth was also fortunate.
Giovanni and his early descendants were very astute bankers and managed their fortunes well. Over the generations the Medici extended their influence by marriages into the royal families of middle Europe.
The huge wealth of the early Medici coincided with the emergence of art and philosophy from the repression of the dark ages. This mix of intellect, sophistication and money made it inevitable that the Medici would sponsor many aspiring talents to do great works.
Artists such as Michelangelo, Leonardo, Donatello and Raphael all benefitted in some way from the Medici influence. Scientists such as Galileo also flourished with the help of the Medici.