Does anyone know the origin of the phrase? Specifically, why does “turn out” imply completion or conclusion?

In a broader context the key to the coherence of any non-elementary set of utterances lies in transitions, and being credulous about those is a sure way to get sold the Brooklyn Bridge. Of course all of us do this all the time without realizing it, to advance our own beliefs about the state of things, and since we usually talk to people who agree with us, we don’t get called on it. Sentences are mostly just lists with descriptors attached to them, and usually the relationships are simple and non-novel enough that if you just pay attention to the nouns in a piece of prose or an utterance, you don’t lose any information. But getting people to accept a novel, specific logical relationship requires transitions, conjunctions and prepositions, the latter if you’re dealing with a physical object or process. To see sloppy transition use in the service of narrative-building, try scrolling through the post-game descriptions on ESPN during March Madness and see the narratives they invent: “The Bruins pulled ahead to a 10 point victory when player XYZ broke Gonzaga’s momentum” – or, listen to end-of-day stock reports on radio news: “Stocks down on concerns about Middle Eastern chaos” (really? You’re sure that’s why?) Post hoc propter hoc is really a special case of this fallacy.

Taleb’s Fooled By Randomness has a nice passage about the abuse of “because” in this same sense.

All I’ll say in the way of a defense of its having those other functions I claimed for it—inevitability, dispassion, an air of discovery, deception, etc., etc.—is to re-emphasize the way in which the use of “it turns out” by lazy writers is parasitic on its use by rigorous straightforward expositors.

Like you said, in the mathematical context it basically means something like “here is a fact that I have discovered to be true, and I’ll suppress the details to keep things moving along—if you wanted to work it all out, you no doubt could”; after seeing the phrase used in this way enough times by honest writers (who actually have done the work they’ve implicitly claimed to), readers might reasonably “let their guard down” when they see it in the future; so if a dishonest writer comes along and says that “it turns out that X”, these readers will take that statement both as evidence for X and for the fact that the writer has done the work to justify X; the trick is that the idea that “when writers say ‘it turns out that X’, X turns out to be true” has been baked into these readers over time.