There's a subtle distinction here between truth-value and semantic-value, one that harkens back to the Russell/Frege debate about 'denotation' and 'sense'. In short, truth-value deals with (bare) facts about the world, while semantic-value deals with impressions of the world. These two often overlap, particularly in analytic and scientific utterances, but they don't overlap by necessity. There was a strong push in the early-to-mid 20th century — peaking with Russell — to ground philosophy in the type of reasoning that the physical sciences use, but by Quine's time that effort had imploded on itself.

As the old joke goes, there are two kinds of analytic philosophers left in the world: those who don't understand Wittgenstein, and those who won't understand him. Ha ha ha…

At any rate, the problem is as follows. We can assign truth-value to any utterance whatsoever based on its conformance to real-world objects or concepts: its capacity to denote. But as often as not that is merely arbitrary, asserting an utterance is false merely because it doesn't point at something we can see or touch. But we assign semantic value in a much more sensitive and context-rich manner. Consider the following utterances:

• The blue door in the garage is locked

This has a specific truth value, and conveys a specific meaning: if true, it means we cannot use the blue door in the garage without the key

• I should have been a pair of ragged claws scuttling across the floors of silent seas

This utterance conveys an emotional state. It's likely true at the time of its utterance (in the sense that it is a true representation of a feeling), but the meaning is something we must dig out.

• Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn

Within the Lovecraft universe this utterance has truth-value and meaning, in the sense that dead Cthulhu is, perhaps, factually dreaming in R'lyeh; in universe it's functionally equivalent to 'the blue door in the garage is locked'. But outside the Lovecraft universe it lacks truth-value or meaning, denoting nothing and conveying no sense to anyone.

• Colourless green ideas sleep furiously

This utterance is tautologically false because the word denotations contradict each other. But it conveys no meaning or impressions. Unlike the 'ragged claws' utterance this is semantically empty.

• Spleznitzia clabulant displa glor.

This utterance is merely grouped strings of letters. It has no truth-value because none of the 'words' have any denotations, and it is semantically empty because it conveys no impressions.

The tricky aspect of this, of course, is that human mind generates meaning through analog processes like metaphor and simile. That's why the 'ragged claws' utterance has meaning; it evokes a particular set of feelings and imagery. With effort, one can generate meanings for the last two utterances through creative association or some such, and once we've created such meanings we can revisit truth-value. But the point is that these are two distinct processes that are too easily confused.

Chomsky observed (Syntactic structures p. 15) that "Colorless green ideas sleep furiously" and "Furiously sleep ideas green colorless" are both "equally nonsensical", but "any speaker of English will recognize that only the former is grammatical". From this we conclude that "the notion 'grammatical' cannot be identified with 'meaningful' or 'significant' in any semantic sense".

The question of semantic interpretation is orthogonal to this point, and we cannot assume any particular view of "semantics" at the time (his semantic theory only started to develop much later). Under a correspondence theory of meaning, neither string compositionally describes a logically-possible state of affairs. However, like many utterances, the string can be pragmatically interpreted as suggesting some other idea, which is not the same as a literal semantic interpretation (it would be part of broader "meaning"), following pragmatic notions developed after Syntactic structures. Semantic content is not just symbol-manipulation, it is about logical relations between contentful concepts and propositions.

It seems to me to wear its truth-conditions on its sleeve: for it to be true, there'd have to be ideas that are colourless, green, and sleep furiously. But these conditions aren't met: ideas aren't green, and since they don't sleep, they certainly can't sleep furiously.

Chomsky might well dispute that ideas aren't green, that ideas don't sleep, and that they can't sleep furiously. These claims are neither true nor false, since they are meaningless. In programmer-speak, expressions like these cannot be evaluated because they have type errors.

Since you mention vacuous truths in the comments, allow me to give my own proof that ideas are green.

For something to be green, it means that if a person with good sight looks at it in good lighting, they experience the colour green. Since an idea cannot be looked at, the antecedent is false, and therefore the statement itself is true. Vacuously, everyone who looks at an idea will experience the colour green! QED.

But now allow me to prove that, in fact, ideas can be looked at. For something to be able to be looked at, it means if a person with good sight faces in its direction with their eyes open and no obstruction, they will see it. Since you cannot face in the direction of an idea, the antecedent is false, and therefore the statement itself is true. Vacuously, everyone who faces in the direction of an idea will see it! QED.

These proofs exhibit a verbal fallacy by saying "X can't be Y" when what we really mean is, "it is meaningless to say X can be Y".

In the sentence "ideas can't be green", the word "can't" really means that there is no sense in which an idea can be green. That is, when you say "ideas can't be green", you are not saying that "ideas are green" is meaningful but false, you are saying "ideas are green" is not meaningful. But then, the negation of a meaningless sentence is also meaningless, not true.