This would seem to result from two processes.
If I translate grammatical structures literally, then the translation will seem foreign or unfamiliar.
An example is if we decided to translate a reflexive verb as a reflexive in English, even if the reflexive is not used that way. We could say "Spanish speaks itself here' to say "se habla español." I would argue that this has nothing to do with the foreignness of Spanish, but with a simple grammatical difference in the way a language divides up its tasks and functions. There is an example from John McWhorter's critique of a translation from the Russian, where the translators use reflexive in English simply because the Russian is reflexive, but it is a simply translation error, in McWhorter's view:
"Russian, like many European languages, often uses self in ways that don’t make literal sense. English only does this a bit: in behave yourself, you might ask just who else you would exert this behaving upon if not yourself. Perjure yourself and repeat yourself are similar. Russian fairly drips with verbs like these, though – you don’t get angry, you anger yourself; you don’t smile – you smile yourself, and so on."
Often Spanish acts that way. We might say "levántate" means "stand up," not "stand yourself up." We use the reflexive to say we eat or drink all of something. "He ate up all the cake." "Se la comió entera." We use a preposition instead of a reflexive for this: eating something up rather than "eating myself it." Falling asleep is dormirse, rather than simply sleeping (dormir.) We don't literally "sleep ourselves." Sometimes the reflexive is just there to make a verb intransitive, or just makes no difference at all, like the difference between morir and morirse. The only real difference is that we tend to say "me muero" when we are speaking hyperbolically: "me muero de aburrimiento." "I'm dying of boredom." We wouldn't say "muero de aburrimiento."
A second way of being "foreign" is by using different registers of the target language. But here an odd thing happens. The foreign elements have nothing to do with the source language at all. Like when a Victorian British translator tries to make Homer sound like Medieval English, because Homer is "old" in a similar way. Here the analogy is forced.
A third way might be from the use of loan words. We can keep a word in Spanish rather than translating it. "Hey, compadre, let's get going." That's fine, but it doesn't take us very far.
In short, I don't really believe that "foreignizing" translation ought to be a thing enjoying a huge privilege over supposedly "domesticating" modes.