As I read him, his approach tends to be straight forward, he's always called it "common sense", and thought the term can be misleading, it's a good way to approach him generally. Understanding and explanation are related.
For him, as a scientist/philosopher, understanding is approached via "methodological naturalism": one studies all aspects of nature the same way, as a biologist studies digestion, so a cognitive scientist studies the mind.
The goal of scientific enquiry is to be able to provide a theoretical account or principle, usually as simple as possible but no less, from which predictions and observed phenomena can be accounted for - under the theory.
On this view, an explanation would be what is predicted from the theory. If the theory of General Relativity predicts that light will bend a certain way given how the sun interacts with a planet, then if the light bends in the predicted way, this counts as an explanation and is understood to follow from the theory.
How, with straight faces, can 'mysterians' even feign any confidence in – let alone understand – their own 'mysterianism — 180 Proof
He cites Hume, Locke and Priestley (among many others) who were wrestling with Newton's discovery of gravity. Prior to Newton, roughly from Descartes until Newton, understanding was taken to mean intuitive understanding: if a billiard ball hits another billiard ball, the cause was a direct contact.
On the old view of understanding, direct contact was
how the world worked. It makes sense "folk psychologically" - to use that term.
Under this view, the material world, was understood as being a big machine, not unlike a clock which could be built by an artisan. The problem was that this manner of intuitive explanation, did not reach the domain of mind, specifically the creative aspect of language use which Descartes thought could not be recreated through an automaton.
Newton essentially believed this view, that the world was a clock-like machine. Only that when he discovered gravity he realized that the universe did not work like a machine, there is no direct contact. As Newton says:
"It is inconceivable, that inanimate brute matter should, without the
mediation of something else, which is not material, operate upon and affect other matter without mutual contact.... is to me so great an absurdity, that
I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it."
By "philosophical", Newton meant what we today call "scientific."
The world does not follow our intuitions. We now strive to get explanatory theories that explain aspects of the world, we no longer seek to understand the world itself, as Chomsky points out in the article.
So by "mysterian" (not a term he likes for himself), or "common sense", he simply means that there are aspects about the world we don't understand, given the creatures we are. Our intuitions mislead us into the nature of the world.