What is the origin of the phrase "turns out"?
Solution 1:
"How it turns out" is also often phrased in the form of, "tell me how it went". "turn" and "went" are directly related, as "went" comes from an old word "wend", which means "turn".
Isn't that interesting? When you ask how something went, you are literally asking how something "turned" out.
Went is the past tense of go. Turn represents just that, rotation or revolution, a thing going.
Solution 2:
Idiomatic phrases like "turn out" just cannot be taken literally, word for word. Over the course of time, the literal meaning has evolved into another layer of meaning that is no longer represented by the interpretation of the individual words in a verbatim phrase.
That is what idoms are: there is a difference between the word-for-word meaning and the meaning of the phrase as a whole. Open any idiomatic phrase dictionary for thousands more examples. As another example, "How are you doing?" has nothing to do with actually "doing" anything either. And if you were doing something, the grammatically correct question should be either "What are you doing?" or "How are you doing this?" But since it is an idiomatic phrase, it just means that somebody enquires after your well-being.
Each idiomatic phrase has its own history, and each will now mean more than the logical sum of their words.
Somewhere in the past, "turn out" probably had something to do with actually "turning" something, or things "going" somewhere, as icnivad has described so well in the previous post.
I can still sense an element of something changing from one state to another state in the phrase "turn out", but to me it is one of many set phrases, with a distinct meaning of its own that cannot be completely deciphered by interpreting the individual words of the phrase.
As a non-native speaker of English, I need to rely on a dictionary that tells me what "turn out" means, compared to the individual meanings of "turn" and "out".
Solution 3:
A baker turns out a cake (or similar) onto a tray from the baking tin after baking in order to cool. The tin is turned and the cake comes out. This is critical moment - a point of assessment for the baker, and so the cake turns out well or badly. Whether this is the origin of the verb I don't know, but it's the most literal and least idiomatic use I can think of, and I'm not even a cook!
Solution 4:
Although I have no definitive answer here, I have an informed one. "Turn out" may come from the metaphor of "destiny or identity is a direction".
The idea of metaphors ruling our thinking and language is proposed by George Lakoff and Mark Turner, particularly in books like Metaphors We Live By and More Than Cool Reason. They present the idea that most figures of speech are based in a metaphorical understanding of the world, metaphors like "life is a journey" (so we use figures of speech such as, "We've come so far", "this is a milestone", "on the road to wealth", etc.).
I would think one metaphor is destiny and therefore identity is direction. Here, to "turn" means to literally change direction, like a compass pointer, and therefore metaphorically to change destiny or identity (of a person or a situation). So if a battle "turns", it changes its destiny. When someone "turns religious", they change their identity or destiny. When milk "turns", it changes its identity for the worse.
When we ask, "How did it turn out?" I think we are calling on this metaphor to say, "What was the identity or destiny of the situation at the end?". If things "turned out well" then they were good; etc. The "out" part may be an additional metaphor that says "out means revealed, in means concealed". So "turned out" means the direction/destiny was revealed.
Notice we also call on this identity or destiny is a direction when we say "how did things wind up?" (wind as in winding road, that is, direction).