Question: In general in English, we don't ever apply the definitive article to languages. We don't say "He speaks the Japanese" or "It was originally written in the French."

But for translated books, they are very often prefaced with a note phrased as *Translated from the Spanish* or *Translated from the Arabic*.

Where does this odd form originate? What is the reason for this grammatical deviation?

Answer: 
“the *adj*” is a reduced form that removes a noun (which is usually obvious from context) because the adjective is what really matters.

In this case, “the Spanish” probably means “the Spanish version”, though there are several other words that would give the same overall meaning.
It's similar to asking the question "What's the Spanish for -something-". For example "What's the Spanish for Supermarket?"

In that case someone is asking for a specific Spanish word (the answer is 'supermercado'). In the case of "Translated from the Spanish" the writer is referring to a specific Spanish text. For example if the quote related to the windmills passage in Don Quixote the English might have the subscript "Translated from the Spanish" where "The Spanish" related to that passage in Don Quixote and not to, say, a guide to the Alhambra.

When we say "Does he speak Spanish?" the question is about the subject's ability to speak (and understand) Spanish generally. This would include the ability to read Don Quixote, understand a sound guide to the Alhambra and to describe a fault with his car to a Spanish mechanic.