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I recently watched an excellent Russian movie "The Horde" in Russian with English subtitles and got totally confused by one scene in it. The scene is as follows. The Khan tells a European ambassador, "Научись, а то тебя женщина твоя любить не будет." The subtitles say: "Take it. Keep trying. Or your woman will not love you." The context is that just before saying this phrase, the Khan demonstrated his superiority over the ambassador in handing a sword.

I am completely puzzled as to how "научись" can mean "take it and keep trying."

My understanding is that "научись" means "get skilled" (in handling a sword in this context). Let me humbly explain why I think so. The infinitive is "научиться," this is возвратный глагол совершенного вида. The core verb is "учить" ("to teach"), and the postfix "ся" converts it to "учиться" ("to teach oneself," that is, "to learn"). "Учиться" is an imperfective verb, and the prefix "на" makes it perfective, "научиться." Since it is a perfective verb, it means that the process of learning is finished, so the meaning of "научиться" must be "to get skilled," and dictionaries seem to confirm this. "Научись" is the imperative of "научиться" and thus must mean, "Get skilled!" I also checked in the Internet how "научись" is used, and my interpretation seems to fit.

And my interpretation of "научись" makes perfect sense in the context: Get skilled in handling a sword, otherwise your woman won't love you. It makes perfect sense because men who cannot protect their women are to be despised. A weak unskilled samurai deserves contempt.

I am totally puzzled as to what I am missing. Being unable to sort it out on my own, I decided to ask you.

My question is this: How can "научись" mean "take it and keep trying," or did I really spot an inexplicable grieve translation mistake in the movie?

Your understanding of word "научись" is good.

I've watched the scene. You've missed a slight pause (or rather that there are two stresses instead of one): it's not "научись", but "на, учись".

"На" is not only a preposition, but also an independent word (not sure what exactly it is). It can be roughly translated as "take it", or "here"

I believe you've figred out "Учись" already. But just in case, it's imperative from "учиться", so literally "go learn [how to do that]", "keep trying" also works.

In this case, "На", as noted above, is a separate grammatical unit. There was a pause with which you can determine that the phrase sounds like "На, учись".

"На" can also be translated as "here you go". But "take it" is quite a viable option.

What I can also state is that "на" is definitely a particle. In Russian, they are used mainly to change the connotation or to add an emotional undertone to the word. In English, the word "well" is a clear example of a particle similar to Russian particles by definition. Again, everything described above applies only to this context. In most other cases, a separate "на" in the written text will be a preposition. The main thing to remember is that "на" surrounded from both sides by punctuation marks will absolutely be a particle.

The topic of particles in Russian is very interesting and not all competent native speakers can identify and correctly separate them with commas in the text. So do not be embarrassed and wish you good luck with it!