Ne pas mâcher ses mots translates literally as “Not chewing one’s words.” The idiom means expressing one’s opinion plainly and bluntly, with no concern for how it’s going to be received. It is equivalent to the (similarly food-oriented) English expression, “Not mincing words.”
This idiom first appeared, albeit in an Old French incarnation, in the thirteenth century. as "ne le querre maschier." which meant being forthcoming about something, not hiding anything - maschier being an old form of mâcher. The idiom reached its modern form in the late nineteenth century.
Not chewing your words implies two things here:
one, that you didn’t take the time to think about (chew) what you were about to say, which might otherwise have led you to edit yourself slightly;
and two, that the words you’re pronouncing have to be swallowed whole, and may therefore be difficult to digest for those on the receiving end.
And this brings us to a related expression: Ne pas digérer quelque chose (not digesting something), which means not being able to get over an incident or something someone said, and being still upset — and possibly holding a grudge — about it.
Le ministre de la défense a la réputation de ne pas mâcher ses mots.
The Defense Secretary has a reputation for strong and blunt answers.
Il avait l'habitude de ne pas mâcher ses mots avec ses collègues.
He was used to telling his co-workers the way it was.