Both the English and American legal systems are based in large part on the writings of Social Contract theorists, like John Locke and Jean-Jacques Rousseau. Legally and politically speaking, this is one of the most influential notions to arise during the Enlightenment.
The United States Constitution and Declaration of Independence draw heavily from the writings of Locke. Large portions of both documents read as if they were quoted directly from some of his works. And the concepts espoused, both implicitly and explicitly, by both documents are backed by concepts he advanced. In particular, things like natural rights, collective sovereignty, religious tolerance, the necessity to impose restraints on the exercise of arbitrary power of the executive, etc.
He even preached specifically in favor of the right to revolution, claiming that because all people have the fundamental rights to life, liberty, and property under the natural law, they could instigate a revolution against the government whenever it verifiably acted against those interests of its citizens. In some cases, Locke even insisted that revolution was an obligation, insofar as it provided a bulwark against tyranny.
This, as you can well imagine, was often cited as a justification for the American Revolution.
If you're interested in reading more about this, the best place to start is with Locke's extremely well-known Two Treatises of Government, paying particular attention to the second treatise, which is his polemic in favor of a civil society based on natural rights. The entire book is even available as a free download (in various formats) from Project Gutenberg.
Others of our laws, particularly those in the juridical realm, like those you cite (intent, burden of proof, no cruel or unusual punishment, jury of your peers, no double jeopardy, etc.) are based on the writings of the French Enlightenment thinker, Montesquieu.
We of course didn't apply everything that he advocated, but there's no denying his strong influence. His The Spirit of the Laws was widely read by the American "founding fathers" and widely disseminated throughout the original British colonies, where they seemed to capture mass appeal because of the growing tensions with England over things like representation in taxation and the right to political liberty.
In particular, James Madison, Alexander Hamilton, and John Jay, in their collected set of writings that were published as The Federalist Papers and argued in favor of the new Constitution, cited Montesquieu at length. Madison famously wrote in The Federalist, number 47:
The oracle who is always consulted and cited on this subject, is the celebrated Montesquieu. If he be not the author of this invaluable precept in the science of politics, he has the merit at least of displaying and recommending it most effectually to the attention of mankind.
He was also very much responsible for what became known as the Separation of Powers doctrine, enshrined in the U.S. Constitution, derived from his commentary on Britain's tripartite constitutional system.
