Why did some Enlightenment thinkers despise democracy?

The Enlightenment was not about "democracy," per se. It was really about "natural law," which meant checks on the power of governments generally, and kings specifically. I can identify roughly three groups of Enlightenment thinkers, only one of which is "democracy" leaning. These include John Locke and Rousseau. At the other end of the spectrum are people like Hobbes, whose idea of good government was "anything better than the state of nature." Voltaire and a third group are probably in the middle.

Thomas Hobbes was perhaps the most pessimistic of Enlightenment thinkers. In his view, unless there were governments, there would be a war of "all against all," and life would be "nasty, brutish and short." In such a world, the main function of government was to keep the peace. In order to do so, it would need a lot of coercive power. Such power (probably in the form of a king) was a necessary evil. As long as it produced a better result than the alternative, anarchy, it should be tolerated. (Later, John Locke would stand Hobbes' idea on its head by argument that a state of nature was still preferable to despotism, in arguing for democracy.)

It is via John Locke that we get our notions of Enlightenment as pro Democracy. (A commenter rightly pointed out that there was an even earlier document, the Dutch Act of Abjuration, that was a forerunner to, and quite likely inspired Locke's work.) The best exposition of Locke's philosophy is found in America's Declaration of Independence which in some places reads almost word for word like a Locke treatise, and could probably have been written by Locke himself. Locke was supported decades later by French thinker, Jean-Jacques Rousseau, who preached the idea of government through "general will," and the exchange of "natural rights for civil rights.

People like Voltaire supported Enlightened Despotism, a middle ground between the absolute despotism of Hobbes and the democracy of Locke, because of the possibility that it could deteriorate into mob rule that would trample on "minority" rights. Instead, despots should be guided by the "humanistic" principles of the Enlightenment. Voltaire drew inspiration from Plato before him.

If you read Plato's Republic, Socrates thought that democracy would lead to dictatorship. Socrates has five levels of society, in descending order of his preference, and one leads to another in that order: 1. Kingship, 2. Timocracy, 3. Oligarchy, 4. Democracy, 5. Dictatorship.

So given that such dominant thinker as Socrates has so great an influence on Enlightenment thinkers, there is no surprise that they would despise democracy.

I would assume the author you are citing means something like direct democracy or selforganizing locally from the ground up. In the history of politics very few people were actually in favour of such a kind of radical democracy (why should they anyway? they would lose their power). Even Robespierre was not for direct democracy but for a representative system with many rules to avoid corruption. The people should only decide politics in the things that don't overwhelm them, for every other thing, people should choose a representative. Many enlightenment thinkers wanted the same. They were in favour of a constitutional monarchy, with the king being a symbol of national unity. More radical forms of democracy were proposed by the first socialist thinkers in England and France (co-operative, economic democracy). Or if you go further back in history to the glorious revolution in England the so called Diggers had radical ideas about what democracy is, but they were ahead of their time (like the levellers of Cromwell's New Model Army).

One must be careful in reading the word "democracy" as it has multiple meanings - and many writers will deliberately conflate and inter-change the meanings with intent to deceive.

One sense is the very literal Athenian democracy - which might be regarded as government by plebiscite. Here every significant public decision is made by a direct appeal to the people assembled, and is nothing more than mob rule. History teaches us that those of influence will make every appeal to the people at the height of emotional intensity, when the ability of the mob to reason is at it's lowest ebb. This is readily seen not only with The Terror barely a decade after Voltaire's death, but repeatedly over the next 250 years during, among others, the Russian Revolution; Mao's Cultural Revolution; and the Khmer Rouge's brief reign of terror.

In contrast to this are various forms of Representative Democracy. The best known examples of this form are the Westminster form developed in the United Kingdom and exported to several Commonwealth Dominions, and Constitutional Republic as developed following the American Revolution and failure of the Continental Congress' Articles of Confederation. The common thread to all Representative Democracy forms is the removal of power from the assembled mob, by having The People elect representatives to in turn perform the duties of government, both legislative and executive.

These flaws of Athenian democracy were well known not only to Enlightenment thinkers but to philosophers as far back as Plato. There is nothing peculiar to the Salem Witch Trials except the ostensible crime. Mobs in all times and all places have, whenever empowered, run rampant over the rights, property, and lives of everyone who opposes, or is believed to oppose, them. Kristallnacht occurred less than a century ago; above are additional more recent examples; and I'm confident the gentle reader can draw from memory numerous examples both more and less recent.

Examining one instance in conclusion, the American Founding Fathers were amongst the most outspoken and noteworthy supporters of "government of the people, by the people, for the people". Yet they deliberately modelled the American Republic's unitary executive after a European Absolute Monarchy "tempered" into the rule of a Platonian Philosopher King, through constraint, by both lack of Judicial and Legislative power and a hard term limit. They had a much higher esteem for George III than they did for his minsters and, rightly or wrongly, felt the need for their rebellion was due more to those ministers than the Monarch himself. They sought to deliberately break the Westminster union of Legislative and Executive authority, in a Prime Minister, into separate roles of President, Speaker of the House, and Vice President (as President of the Senate). That last rapidly proved unworkable, but the former remains today a key constraint on the President's executive authority.

So in the referenced quote it is not "representative democracy" that is despised but "Athenian democracy". The so-called Absolute Monarchies of Europe never were absolute. Those monarchs had allied themselves with the burgeoning middle class, who accepted increased royal taxation as the price to be paid for increased royal power. In exchange, those monarchs introduced Rule of Law that freed the middle class from arbitrary action by the nobility. This "enlightened absolute monarchy" is what Voltaire and others are espousing, which the American Founders attempted to cut from whole cloth, and which the Westminster systems have evolved into.

And as a final note: Men of thoughts and letters, of not just the Enlightenment but times previous and since, quite rightly in this author's opinion, have regarded "Athenian democracy" not as the triumph of Rule of Law; but rather as the triumph of its antithesis: unbridled tyranny, the tyranny of the mob: with the strength of its greatest member, times thousands, and the intellect of its feeblest, divided by thousands.