What's the meaning of "let it fly" in this context?

Your confusion about "let it fly" is entirely justified; the "it" in "lets it fly" could refer to either the dove or the bullet. However, the additional context of "Nate... cradles his weapon" removes the ambiguity: cradling your weapon means "holding it in a manner similar to how you would hold a baby". (Generally I would consider this position to be holding the weapon diagonally across your body, with the stock on or near your hip and the barrel pointing toward the sky from the opposite shoulder; you would have one hand on the firing grip (but NOT with a finger on the trigger!) and the other on the barrel grip.¹ )

Since this is not a position that you can target and fire from, the only option then is that he let the bird fly (away). And he lets it fly away because it is at the extreme end of his range, and he would have to be lucky to actually hit it.

The writer could, and probably should, have avoided the ambiguity by saying "lets it fly away" or "lets the dove fly".

¹ _{There are other cradling positions, but none of them involve holding the baby's feet up to your shoulder and pointing its head at a particular target.}

It could be that "let it fly" is like the Nike slogan "Just do it"--suggesting not holding back, taking risks. "He lets it fly" could mean he fires the gun--he takes the shot, without being able to aim precisely, because the dove is just out of range. Therefore he has to be lucky if the shot is to succeed. But if you read the NYT article in full (nothing like context!), the meaning is clearer. The dove flies away and, like college for Nate, is just out of sight/reach; he will have to be lucky to get there. Apparently cradling the shotgun is the opposite of raising it, aiming, firing.

First of all, I think @Hellion's answer is correct (so I will not be expounding on the importance of the word cradle) and I want to add to that.

A dove hurtles out of the cedar trees, but Nate Triggs, in camouflage vest and cowboy boots, cradles his Mossberg shotgun and lets it fly.

If we breakdown the first sentence we identify that the author talks about a dove which hurtles out (flies fast out of trees) suddenly from the trees. I think the word hurtles is also key as it means moving very fast in a particular direction suddenly. The author almost means as if "A dove suddenly appears from the trees" and Nate can only cradle his weapon because in the mean time the bird flies out of his range to shoot it with any real chance of hitting it. He had a small window to try shooting it and he chose not to as there was little likelihood of being successful.

Moreover, if you simply read the next lines from the link of the article you have provided, it says:

Like the dove, college feels borderline out of reach to Nate too. He gets conflicting signals from home and school about whether it’s even a good idea.

I think this passage is immensely important and the OP should have included it in the question. This clears up any ambiguity on the author's part. Here, the author says - "Like the dove" which is out of reach of Nate (and consequently can't hit it with a shot), the college is also almost out of reach too (can't have a real shot at it either). Thus, this implies that the dove flew away. This passage clears everything up.

Anyway a picture of Cradle Carry from Washington Hunter Ed Course. It is kind of self-evident why it is not a firing position.

The passage is poorly worded. I don't think you would have seen it in the New York Times 30 years ago, but our language is becoming degraded, and even at the New York Times we are now seeing some strange constructions. The writer, in my opinion, should not have used the expression "let it fly" here because of its ambiguity (let the bullet fly, or let the bird fly?) Or he could have written, "let it fly off," which would have clarified things immensely. And yes, I agree that he meant that he let the bird fly away, as it was too hard a shot.

Update: Thank you so much, friends, for all of your generous replies!

One of my classmates ( we are a group of online English learners) emailed the author asking these questions (we've been discussing these a lot, like we are doing here, and haven't reached a consensus ) . The author writes back, and here is her reply ( most of you have got the right understanding, according to her) :

...So in answer to your questions:

1. "It" is the dove, not the shotgun, so Nate is letting the dove fly away rather than trying to shoot it. I can see how there might be some grammatical ambiguity here, since "it" comes after "shotgun," and could be read as an echo. But I think most native English speakers would intuitively understand that "it" refers to the dove.

Here at the Times, we are often admonished to be conversational. The sentence as I have written it is, I think, more conversational than if I had repeated the subject and written:

A dove hurtles out of the cedar trees, but Nate Triggs, in camouflage vest and cowboy boots, cradles his Mossberg shotgun and lets the dove fly.

2. Nate did not shoot the dove because he thought it was too far away, and he would miss. So he used his judgement, and did not even try. He told me that he would have to be lucky to kill the dove at such a distance.

I used this thought as a kind of a metaphor for how he felt about applying to college. At that point, college seemed to him a very distant possibility, not very realistic, so distant that perhaps he should not even try. This idea foreshadows the ending of his story, because in the end, he decides not to aim for college, but to take a job as an auto mechanic instead, at least for now.

By the way, Nate and I went hunting with his friend, Tyler. Tyler did try to shoot the dove, and he missed. This told me that Nate had better judgement than Tyler. So I am optimistic that Nate will go to college in the future. I believe in his good judgement.

Again, thank you guys very much! I have read all of your answers and they are very helpful. I'm just a beginner and I will try to ask more questions in this wonderful forum! Many thanks.