Words like “relax” (rirakkusu) and “hose” (hosu) seem like something that would have existed before contact with English-speaking people.
This is true in a lot of languages and I think part of the reason is just the ever presence of English in media and music – words become familiar and comfortable to use in given situations, even if Japanese (or polish, or Hungarian or whatever) words already fulfill the lexical gap
The words often have slightly different connotations. For example, the English loan word *kyanseru* (cancel) is an informal, modern-sounding way to talk about canceling something. If you were talking about canceling a contract or other more formal arrangement, you would likely instead use the Chinese loan word *kaiyaku*. And if you wanted to talk about canceling in a more general or abstract sense, like taking back your words, you might instead use the native Japanese word *torikesi*.
Japanese is not unique in this regard. English has many French and Latin loan words for terms that already had native semi-equivalents: pain/hurt, rage/anger, response/answer, prior/before, commence/begin, creed/belief, abdomen/belly, corpse/body, fraternal/brotherly, construct/build, etc.
While the other answers are also correct about there sometimes being subtle differences between English and Japanese words, there’s also another major historical/cultural factor: English is “cool” (*kakoii*) in Japan.
While other languages certainly borrow loan words, this is a uniquely Japanese phenomena related to the country’s post-WWII relationship with America. Many people forget that while we did bomb the hell out of them in WWII, we also helped build the country back up afterwards (and secretly forced them to adopt modern rights for women and some other progressive stuff). This it led to a unique love of America in Japan: there’s a reason why the only non-English loan word I can think of is bread or “pan” (it’s use predates WWII).
As an example, here’s a story my Japanese teacher told me. She was in Japan having dinner at a restaurant with an older Japanese woman, and the woman ordered *torii-niku* (chicken). The younger waitress “corrected” her and said “you want to order the “*tchi-ken*”?’
The older woman corrected her back “no, I don’t want to order the *tchi-ken*, I want to order the *torii-niku*!” Eventually the waitress gave up, but it highlighted the basic fact that in certain cases in Japanese the English versions of words are “hip”, and what the cool young people use. It’s also used heavily in advertisements, apparel (see http://www.engrish.com) and (evidently) trendy restaurant menus. But it’s not like the original Japanese words went away or anything, and (for instance) older people still use them exclusively.
If you watch anime or otherwise listen to Japanese pop music, you’ll hear English words used this way frequently. It’s not all English words though, because not all Japanese (young or otherwise) know English well enough to have a full vocabulary. It’s just a certain subset of words that the “cool kids” use that everyone (or at least the younger generation) knows.
Same with hindi/urdu unfortunately amongst many younger gens. Its usually just to sound “wake”. Many developing countries in the world, how much ever they act like Patriots have a hell lot of inferiority complex towards Western countries.
Because Japanese students learn limited English over the course of their education, it’s easy for loan words to cross into the lexicon. You will see, sometimes, these loan words will become more common than the original Japanese, and sometimes they are only used in special situations like music.
Relax, for example, can be リラックス (rirakkusu) or it can be 寛ぐ (kutsurogu), “to feel at home” or 寛げる (kutsurogeru), “to loosen or ease”, or 憩う (Ikou) “to rest”
Hose can be ホース (hosu), or it can be 導管 (Dokan) meaning “Conduit”