In 17th Century Europe, philosophers realized that Galileo was right about the universe. It did seem to be governed by "natural" laws that did not require supernatural intervention. Even though this was not called the Age of Enlightenment until the 1770's, my own personal date for its beginning is with René Descartes and the publication of his Principles of Philosophy in 1644.
The short-short version of Cartesian philosophy is this: People can use reason -- the ability to think about things -- to learn truth. Aristotle had asserted that experience was the only valid teacher, and all truth could be derived from it. Descartes said that some things were accessible only through reason, but they are still true. He also affirmed that natural laws could govern the world without help from supernatural sources. Finally, Descartes is responsible for the "mind-body split" that divides the world into material objects and the mind that thinks about them. "Mind," for Descartes, is close to being what "God" is to William Bradford.
In the Cartesian universe, everything has been wound up, presumably by God, and set in motion according to natural laws. Within that system, people have about as much free will as they have under John Calvin's predestination, but it's going to open the door for Deism, later. Oh, how nervous this made the Puritans. So, they ignored Descartes, and hoped he'd go away.
John Locke, the most important British philosopher ever, was impossible to ignore. His ideas fueled the American and the French revolutions, and still resonate with us today. They underlie our government, and were largely responsible for the shift away from Aristotelian logic.
Locke's philosophy, like that of Descartes, can't be compressed very easily into little digestible bits. But that's not going to stop me from trying. Just remember that this is "Locke Light." The real thing is a lot more complicated.
Locke was a utilitarian; he defined the "moral" thing as that which would do the greatest good for the greatest number of people. Many governments operate on this principle, and it beats the heck out of governments who only want to do good to the six guys who have all the power. Next, Locke explained that all men (not women, yet), in their "natural" state, are equal. In other words, all this "blue-blood" stuff about aristocrats being a higher order of human is just wishful thinking. Bad news for monarchs, who made a good living by convincing people that they were special and should not be messed with.
These equal men have the natural right to own property. Men have other natural rights -- life, freedom, and health. Sound familiar? It should. A just government, Locke said, is one that supports these natural rights and then gets out of the way.
To support the idea of natural rights, which people are born with, Locke goes into a complex argument about where they come from. It doesn't involve God, but instead affirms the inherent worth of human beings, thereby opening the door for later Humanists. Locke's idea, which now pretty much permeates Western thought, is that humans are born as blank slates (his literal tabula rasa), onto which experience and reason write character, personality, and life choices. This is much more appealing than the Puritan idea that humans are born as disgusting, sinful beings, destined for hell, so, naturally, it caught on.
Weirdly enough, Locke profoundly influenced a man with whom he otherwise had little in common -- the architect of the Great Awakening, Jonathan Edwards. Edwards still believed in original sin, but he couldn't quite lose the idea that Locke was right -- people are made, not born.