I haven't really settled on a firm answer for this for myself, but here are some possibilities.
Her high rank and social status protected her. As Thomas Dudley's daughter and Simon Bradstreet's wife, she was the sort that the authorities -- the rest of the authorities, that is -- would think twice about lowering the boom upon.
She wasn't actually doing anything so radical. Maybe she "got away with it" because there was nothing she was really getting away with. Maybe the people of her time did not give a collective, loud gasp when she said hey, folks, women have talent, too. Likely women chiding men was acceptable sport in that day, just as it is in our day; or as those under authority these days chide those who have authority. You've all read "Dilbert" so you know what I'm talking about; we do what the boss-man says, but talk privately about just how pointy-haired s/he can be at times, and that is perfectly acceptable. In the seventeenth century, the lines of authority were much tighter, and something as disrespectful as "Dilbert" would never fly. But whenever we paint an era only in black and white, we get into trouble. English women had an inferior legal and social status at this time, but they were not slaves; they made their will known, and they pushed back.
I think this is the best explanation. It jives nicely with one of my "first principles" of historical interpretation: always assume first that what people in the past are doing is NOT radical or ground-breaking, even if it looks like it at first glance. Often we just don't understand the time or the motivations, and (more important) we have a cultural bias of our own that makes us identify everybody we can as a trailblazer or a radical because we just love superheroes so much. Personally, I don't think Anne had a radical bone in her body. She wasn't made to be a rebel. She may have been spunky or feisty, but rebellious? Not the way I read her.
I think too we need to remember she was not writing to make a public statement. Her poems were for herself and for those in her circle. She didn't plan on The World reading her stuff. And further, let's keep in mind that Englishmen were familiar with the idea that some women were exceptionally talented, the supreme example there being Queen Elizabeth.
The really good question in all of this is, why did her menfolk take such pains to get her work published? This is the part of the story that I will go ahead and say is -- okay -- radical. Sort of. (Thomas Dudley in a Superman suit -- if you knew the man, you'd realize how funny that is.) They would not have done it to bring fame to Anne, never to "get her out there" and "let her talent be seen." Puritans would have regarded such motivations as worldly arrogance and vanity. Instead, they must have seen her poetry as having some role in their life's work, which was to bring the Kingdom of God to earth, and as much as possible, to build a truly godly society in New England.
Hey, I really like that. A woman writing classical-esque poetry based on the science and philosophy of the day, the sort of poetry that usually only men wrote -- and these puritans saw that as, in its own way, having a role in their grand experiment of building a truly godly society. More food for thought in that, eh? But I'm going to leave this particular line of discussion and move on to something else next time.