Why are seemingly foreign words such as hors d’œuvres, maître d’, garçon, and Gesundheit used in American vernacular?
All languages in the world have borrowed words from other languages. That is how language works. In the title to your question there are four words of non-English origin: "foreign", "used", "modern" and "vernacular".
There are words that are retained in their original language because they impart a certain feel, say, of elegance, snobbery, belonging, etc. hors d' oeuvres, maitre d', and garçon were retained by French restaurants, and came to be generalized because of the connotation of elegance and sophistication of French restaurants. The same could be said of concierge in a hotel.
There are words that perfectly capture a situation in a foreign language, which cannot be captured so precisely in English. An example of that would be schadenfreude (the pleasure derived from the misfortunes of others.) English 'borrows' them.
There are words that have been carried over from waves of immigrants, words that were used enough by a group, or captured the fancy of other English speaking people, that they stayed in the language. A typical example of this is the story of the Pennsylvania Dutch.
"German immigrants came to Pennsylvania in 1683 and settled in the appropriately-named “Germantown,” now a section of Philadelphia. As Pennsylvania grew, more Germans came to the colony and more them pushed further inland, settling in Berks and Lehigh counties in the 17th century..."
We inherited German words like Verboten, or Gott in Himmel (the im having been changed to in), and they have adopted English into their dialect: outen the lights is a common expression around the Lancaster area of Pennsylvania, spoken by non-Deutch people.
The reasons are varied, but there are several that are most prominent. I've listed three.
This is mostly anecdotal (I can't find/don't have/remember my sources):
English as a language can be limiting. Compared to another language, say German, we simply don't have the fluidity of vocabulary to make up words. The common perception is that the German language lends itself to fairly easily building of words, some a portmanteau of pre-exisiting words. This is how we find ourselves borrowing a schadenfreude, because we simply don't have a single word to express the concept. It's literal translation is Harm-Joy, hardly an original construction and it barely makes sense, considering what we use it for. If we were to make an identical construction in the english language, it'd sound silly.
The use of latin (and descendant words) evolved from the need of two estates (clergy and nobility) to distinguish themselves from the filthy 3rd estate, the commoners. See, back in the middle ages, English language (and Germanic languages in general) was for the use of the unwashed masses. French and Latin were the preference of nobility and in that vein, they refused to address many concepts using the language of commoners, instead choosing to borrow from Latin. This is how you'll find the word horse in colloquial use and equine in more academic references. This largely accounts for the large number of French and Latin loan words and root words we have in english.
EDIT: I found something in the way of a reference, but it doesn't corroborate exactly what I have here: http://www.uta.edu/english/tim/courses/4301w99/lge.html