First, it should be clear that in natural languages nothing has a rigid definition. To make sense of a sentence is thus to select "definitions" so that the sentence has the intended meaning. This is done through common sense and context.
A core principle of communication is to be relevant. A obvious tautology defies this principle, as there is no information conveyed. Whoever interprets the semantics of a sentence to be a tautology usually just discards this interpretation, as it normally turns out to be just a misunderstanding.
If the intended meaning is a tautologies, they, e.g. unreal conditions, are always marked ("If I were a boy").
When confronted with "colorless green ideas sleep furiously" the "logician's" meaning is immediately discarded, and the hearer looks for increasingly unrealistic interpretations of the words, until they eventually will give up, since without context, this sentence just can't be interpreted in a sensible way.
Without context, it is impossible to decide whether the meaning is "all colorless green ideas sleep furiously" or "there are colorless green ideas that sleep furiously" or "the way colorless green ideas sleep is (always?/usually?/sometimes?) furiously.". If a quantifier is a ∀ or a ∃ is usually decided by context. And if green ideas can't be colorless, then the above sentences are either right or wrong depending on the quantifier.
That is what he meant with nonsensical: There is just no obvious parsing into a logical statement.
Side note: When delving into metaphors, one could almost use this sentence: There are good ideas, let's call them "green", and bad ideas, let's call them "red"... Some ideas don't have an obvious color you can see; they are colorless. But what if they were, in reality, green? Colorless green ideas sleep furiously, annoyed at how you don't invoke them because you can't see their value. (in this contrived example, the sentence is not a tautology)