The original meaning of "intelligible" was of that apprehended by the intellect as opposed to the senses, for example Plato (in Latin translations) talks about intelligibles (forms) vs sensibles (things), see Karl's Plato’s Two World Theory, and Aristotle's use is similar.
In more recent philosophy there are two somewhat different meanings of "intelligible", both distinct from the traditional one (although inspired by it). For example, Creighton in The Form of Philosophical Intelligibility writes:
"We habitually assign to philosophy the task of "explaining" the world, or of rendering experience "intelligible"... It would seem necessary to understand as clearly and definitely as possible what type of explanation philosophy may properly be expected to furnish before any discussion is in order regarding its competency to fulfill its task, or concerning the relative value and pertinency of various systems."
This reading links intelligibility to explanation, and on explanation there is of course also a long philosophical tradition. Aristotle's four causes were originally construed as four ways to answer "why" questions, i.e. they are more properly translated as reasons or explanations. Only one of them, the efficient cause under which mechanistic explanations fall, was retained by modern science as fundamental, and eventually "cause" came to be identified with what Aristotle called "efficient cause", with additional twist that such causes are also assumed to be governed by (often deterministic) laws. Modern philosophical naturalism largely adopted this outlook, but with a revision. Even in modern physics mechanistic explanations are insufficient, Chomsky's view was abandoned there along with classical mechanics, and replaced with a more general paradigm of (efficiently) causal explanation. On the other hand, this is perhaps why many people still see quantum physics as not sufficiently intelligible.
Evolutionary explanations in biology are often teleological rather than causal (physiological traits are explained by being advantageous in species' environment, for example), corresponding to Aristotle's final causes, but such explanations are usually seen as derivative from efficient ones via the mechanisms of mutation and natural selection. The fundamental priority of causal explanations depends, however, on the success of the reductionist/physicalist programme, which in philosophy of mind is controversial, see Is there a causal influence of the mental on the physical? Modern philosophical theories of scientific explanation are well-covered by both SEP and IEP. The Friedman-Kitcher unificationist theory is perhaps the most popular, in Kitcher's summary:
"Science advances our understanding of nature by showing us how to derive descriptions of many phenomena, using the same pattern of derivation again and again, and in demonstrating this, it teaches us how to reduce the number of facts we have to accept as ultimate."
A somewhat different meaning of "intelligibility" puts emphasis not on explanation but on meaningfulness (the two are obviously closely related), "intelligible" is that which meets the conditions of the possibility of being meaningful. Traditionally, such conditions would include logical laws, like identity and non-contradiction, famously challenged by Hegel, and on some accounts even the principle of sufficient reason, although its inclusion is rare today. The prominence of this view can be traced to Kant's "conditions of the possibility" of experience or knowledge, and his employment of these notions in transcendental arguments.
Aside from the positive use, where such conditions help analyze the structure of our cognitive faculties, our "intellect", Kant effectively coined a new type of philosophical argumentation, where a view is argued to be not even false, but worse, unintelligible, because its notions violate some conditions of meaningfulness. For instance, according to Kant, the notions of traditional metaphysics are unintelligible because they attempt to apply categories of experience beyond any possible experience, to things in themselves. Such approach was since used by many prominent philosophers whose positions are very far from Kant's. For example, Wittgenstein argued against private sensations and languages because such things have no criterion of correctness required for their intelligibility, see Did Wittgenstein consider the possibility of a private language with public content? Similarly, Putnam in Realism and Reason criticized (metaphysical) realism positing "mind-independent" reality expressed in mind's concepts that "correspond" to it as unintelligible:
"But it is unintelligible, from my point of view, how the sort of relation the metaphysical realist envisages as holding between a sign and its object can be singled out either by holding up the sign itself [Cow], – or by holding up yet another sign, thus [Refers] – or perhaps – [Causes]... One of the puzzling things about the metaphysical realist picture is that it makes it unintelligible how there can be a priori truths, even contextual ones, even as a (possibly unreachable) limit."
