This is closely related to the economic concept of the "Invisible Hand"[0] and also explains why planned economies never seem to work, no matter how well-intentioned they are (see the current state of Venezuela - although in their case, as in almost every case of central planned economies, greed and corruption were the prevailing forces of the ruling agency)
"Invisible Hand" turns out to be just another word for a feedback loop. Something we've learned quite a lot about in the past century. Feedback control is not magic. It does some things right, and it dumbly fails in some other cases. It's more resilient and tamper-proof - it's close to impossible for a single individual to control this system. That's why people like it. But it's also much less efficient than central-planned economies could be. In a way, running economy by self-cancelling feedback loops is outsourcing computation to the physical world. Instead of figuring out - using brains or computers - how much widgets a factory should make, you just let people buy and sell and get rich and go bankrupt, until the system spits out a stable state. Very resilient, but very wasteful.
I think central planning is much too fragile for today's world. But who knows, it may be ok for tomorrow's. I believe that prerequisites for successful central planning are having a benevolent planner backed up by huge computing power, so that the planner can respond to changes everywhere promptly enough. A friendly AI, if you like.
The "invisible hand" metaphor used by Adam Smith is not an explanatory mechanism, and if anything is an admission that the specific mechanism isn't understood. In use at the time and earlier, it had the sense of "the invisible hand of Providence" (or God). Though Smith, as Hume, was almost certainly what we'd now call an athiest.
He used the term three times, in three different books: The Theory of Moral Sentiments, then An Inquiry Into the Nature and Causes of the Wealth of Nations, and finally in a book on the history of astronomy. It's clear from context that Smith wasn't embuing markets especially with invisible handedness, but using a common phrase of the age.
The modern invention of this metaphor dates to the 1930s and 1940s, being first used in its modern sense by Paul Samuelson, and latched onto like a desperate child by the budding organs of the Mont Pelerin Society, better known as the von Mises / Hayek / Friedman / Rothbardian variant of Libertarian theology. Its popular significance grew after publishing of Adam Smith's Invisible Hand, a compilation of modern economic fallacies miscast as truths, by Regenry Press, a Libertarian propaganda mill, in 1963. You can trace the evolution of the term via Google's Ngram viewer.
One of the more notable "quotations" from Smith's Wealth of Nations
Economic historian Gavin Kennedy has traced this history in depth, published multiple papers on it, and writes a blog, "Adam Smith's Lost Legacy", which I highly recommend.
(You'll also find some discussion of the false myth that's developed over the term in the very Wikipedia article you've linked.)
Also, Adam Smith's example of the invisible hand was as a mechanism to prevent so-called free trade, and raise protectionist tariffs. The people you mentioned have twisted Smith's words into the exact opposite meaning - they say Smith's invisible hand sweeps away protectionist tariffs and allows international free trade. He said the complete opposite of what they say he said.
Thanks, I was going to mention the context of use but needed to go back to confirm what the usage was.
I am aware that his one use of "free market" was in a passage describing protectionist trade practices favouring the woolens manufacture industry in England: keeping raw wool import costs low and preventing import of finished goods, thereby maximising the revenue-cost differential, which is to say, profits.
Here's the beginning of that section, which generally inveighs against protectionist retraints on trade, most especially the "Corn Laws" -- limitations on grain imports to England:
By restraining, either by high duties or by absolute prohibitions, the importation of such goods from foreign countries as can be produced at home, the monopoly of the home market is more or less secured to the domestic industry employed in producing them. Thus the prohibition of importing either live cattle or salt provisions from foreign countries secures to the graziers of Great Britain the monopoly of the home market for butcher's meat. The high duties upon the importation of corn, which in times of moderate plenty amount to a prohibition, give a like advantage to the growers of that commodity. The prohibition of the importation of foreign woollens is equally favourable to the woollen manufacturers. The silk manufacture, though altogether employed upon foreign materials, has lately obtained the same advantage. The linen manufacture has not yet obtained it, but is making great strides towards it. Many other sorts of manufacturers have, in the same manner, obtained in Great Britain, either altogether or very nearly, a monopoly against their countrymen. The variety of goods of which the importation into Great Britain is prohibited, either absolutely, or under certain circumstances, greatly exceeds what can easily be suspected by those who are not well acquainted with the laws of the customs.
That this monopoly of the home market frequently gives great encouragement to that particular species of industry which enjoys it, and frequently turns towards that employment a greater share of both the labour and stock of the society than would otherwise have gone to it, cannot be doubted. But whether it tends either to increase the general industry of the society, or to give it the most advantageous direction, is not, perhaps, altogether so evident.
The restraint of government intervention (here as so often elsewhere noted in Wealth) is by government, yes, but quite clearly on behalf of specific powerful commercial interests. It's that* power Smith is hoping to curb -- directly then a restraint on excessive power accumulation by means of commerce, trade, and manufacture.
[0] https://en.wikipedia.org/wiki/Invisible_hand