The Quaker faith was started in the mid-1600s, beginning from the time in 1652 when George Fox (the founder of the religion) felt that he was called by God to found a new religion. The founding of Quakerism was made possible by the same sorts of impulses that drove the Reformation in general.
One of the major ideas of the Reformation was that people should not have to rely on priests and bishops to tell them what God wanted. This was why Luther, for example, wanted all people to read the Bible on their own. Fox took this idea one step further. He argued that there should not even be pastors who would deliver sermons. Instead, each person should listen to the "Inner Light," or voice of God, and speak to the others as he or she was moved to do.
So, in answer to the question of why Quakerism was founded, you could give two answers:
- You could say it was founded because Fox felt called to found it.
- You could say it was founded because people wanted to control their own relationships with God and because people felt that they were able to understand God's message without being told what that message was by and priest or even any pastor.