Two major reasons for this : The introduction of printing through Johannes Gutenberg, and the Reformation which implied that every Christian should be able to read the Bible, which made reading accessible and interesting. The Reformation spread first among craftsmen and merchants who could read.
The progressive introduction of paper in the 14th, 15th and 16th century (17th in England) also made writing much cheaper (parchment was made of fine leather and was very expensive).
Most peasants in the medieval times could not read or write, even with basic education (Joan of Arc, born 1412, daughter of a rich peasant or landowner, could hardly write her name). One reason for this was the absence of a real written culture in the spoken languages : clerks and intellectuals wrote mainly in Latin, and merchants used the so-called "lingua franca" (or its equivalent in the Baltic sea) which was a pidgin of Latin, early French, and viking languages (old norse).
The 16th century is the real beginning of alphabetisation, and public schools (run by the community or the state) began to spread. In France, the 1539 Ordinance of Villers-Cotterêts marks the beginning of the French language, but it is also a sign of the progress of written culture in France, since it is the last of a series of royal laws that aimed to replace the Latin language with spoken languages (mother tongues) in official documents.
The vast majority of schools in the modern times were run by churches, cathedrals (there had been cathedral schools since Charlemagne), local parishes, and monasteries. There is a debate concerning the role of (educated) women and mothers in the education of children.
In the rich and noble families, preceptors were hired (this role could also be fulfilled by the local chaplain) or the kids were sent to some elite schools (monasteries, schools of the Society of Jesus...). The schools were mostly run by various religous institutions.
It's worth noticing that some countries were much more educated than others : for example Switzerland, especially in Geneva and its surroundings.