A dig is a "a criticism or insult that is directed toward a particular person or group."

Subtle means "hard to notice or see; not obvious."

So this is "a criticism or insult" that is "not very hard to notice" or "is obvious". This phrase uses understatement to soften the effect a little - saying "an insult that is hard to miss" is more direct and stronger than "a not-too-subtle dig". The phrase a not-too-subtle dig is a common one. Some examples:

...said Walmart in a news release, in a not-too-subtle dig at Amazon.

The first part of the quotation contains a not-too-subtle dig at A.P. Watt.

It was a not-too-subtle dig at Mrs. Beaufort, but no one seemed to connect the remark to her.

No . I do not know who first used this locution, but it is relatively recent. It is meant to jar on the ear as a way of grabbing attention. t is not a common mannerism used in talking about someone's personal or professional deficiencies.

So the adjective is being used as a noun object of 'do'. In your example, the speaker is saying that the person in question entirely lacks subtlety in what s/he says. S/he is probably completely tactless and doesn't 'do tact' either.

You can view it, in its origins, at least, as a sort of figurative usage of the verb 'do'. But it has always had this stretched potential. We might speak of somebody coming in to do the carpets, where what is meant is that s/he is a carpet cleaner. Margaret Thatcher said that she thought it would be possible to "do business" with President Gorbachev. So the feature you are asking about is no more than an extension of an already established freedom.

For some reason this usage has not managed to sneak into the Cambridge online Dictionary. But Collins does idiom (which is getting closer to your example) much more fully.

16. VERB You can use do to say that you are able or unable to behave in a particular way. [informal] 'Can't you be nicer to your sister?'—'Nice? I don't do nice'. [VERB adjective]

It describes the usage as "informal", no doubt because it is hard otherwise to account for the use of an adjective as the object of a transitive verb. In fact, though, there is a precedent for it in ancient Greek and Latin, where the neuter form of adjectives could stand for nouns. English schoolboys would think of this a implying an unspoken noun which the adjective qualified. So our first attempts at translation would be: "I don't do a nice thing (or things of the neuter plural was used)" So there is no reason why it could not be used poetically or rhetorically for emphasis or effect. I could even imagine a letter of application from an extremely confident applicant trying to make an impression in that way, despite the risk of irritating the old buffer that might be reading it.

Many, if not most of the contributors to English Language and Usage are native speakers so we have spent a lifetime acquiring our vocabularies.

As to how we learn the subtleties of meaning one of the ways is by asking the people from whom we learned them, often our parents, but also from reading and hearing the words in context.

For instance we know that "replete with" means "full of" literally in the sense of being full of food but sometimes means other things metaphorically while "fraught with" means "full of" in the sense of being encumbered because we have read many sentences using both words where the context is clear. We know that a bucket cannot be either "fraught" with water or "replete" with water because we have not read sentences with that construction. If we read one we would know that its use was either erroneous or poetic.

The way you, as a student of English, can learn these subtleties is to read and listen to as much good English as you can and, when you find a word you either do not recognise or of the subtle meaning of which you are unsure, to think about it and compare it with other contexts in which you seen the same word.