The idea that democracy is ordained by the laws of nature goes back to the political thinkers of the Enlightenment. These were people like John Locke whose ideas had a tremendous influence on the thinking of the American colonists at the time of the American Revolution. What this statement means is that all human beings have been given certain rights by nature (or God) and that democracy is the best way for people to enjoy those rights.
According to Enlightenment thinkers, all people have certain rights by virtue of the fact that they are human. According to the Declaration of Independence, people are “endowed by their creator” with these rights. These are things like the right to their lives, to their liberty, and to their property. In other words, God/nature (Enlightenment thinkers sometimes used these ideas interchangeably) wants people to be safe from being killed, to be able to own property without fear of having it taken away, and to be able to exercise certain liberties.
Enlightenment thinkers also held that democracy was the best way to protect these rights. They felt that a good government was one which protected those rights. The believed that democracies would be less likely to allow people to be killed, dispossessed of property, or deprived of their rights by the government.
Therefore, since nature/God wants people to have these rights, and since democracy is the best way to secure these rights, nature/God has ordained democracy as the proper form of government.