There is some snobbery (or pretentiousness) connected with the nobiliary particle, rightfully borne or fraudulently assumed. Here is an astute (and amusingly cynical) observation that I saw on a web site (see attribution below), from the 19th century, which nevertheless still rings true today: "We came across this quotation in Philip Gilbert Hamerton’s book French and English: A Comparison(1889): 'After careful observation I have arrived at the conclusion that the French de  before a name, whether rightly or fraudulently borne (for that makes little perceptible difference), is equivalent to about ten thousand pounds in the [London] marriage market and will often count for more. It is wonderful that it should be so, considering that all French people know how frequently the de is assumed; but it seems to be valued as a mark that the bearer belongs to the gentry, which, in fact, he generally does. The genuine nobility who have become too poor to keep a place in genteel society, and have to work for their living, seldom retain the particule, or retain it only for a short time. If they did not drop it themselves the world would drop it for them.' (quoted from http://www.grammarphobia.com/blog/2010/07/nobiliary-particle.html) In every age, there are the genealogy snobs, the nouveau riche and the nouveau pauvre (distressed royals, aristocrats and gentry).

The short answer is that it could be (1) a nobiliary particle, denoting nobility, or (2) a preposition denoting a place of residence, or (3) just something that people add to their own names for reasons of their own. The long answer is here: https://en.m.wikipedia.org/wiki/Nobiliary_particle

Regarding Nicolas de Staël, he was born Nikolai Vladimirovich Stael von Holstein in Russia. The Staël von Holstein family were originally Danish barons. One branch moved to Russia. Holstein is now in Germany but it was previously variously part of Denmark and part of the Holy Roman Empire. In short, Nicolas de Staël was a minor émigré Danish nobleman born in Russia. The Nicolas de Staël name was one that he have himself so that he would sound aristocratic and French and not Russian, Danish and German.

In Dutch 'de' is a definite article, 'the', so the meaning is different from that of French 'de' ( meaning 'of' or 'from'). In the Netherlands surnames with 'de' are quite common, e.g. 'de Groot' ( meaning: the Great), originally an apposition to the first name (as in King Alfred the Great, he was a great king). Dutch names with 'de' often denote a colour, especially 'de Bruin' (the Brown), 'de Groen (the Green), de Wit (the White), de Zwart ( the Black) etc. In names denoting that a family is originally from a certain place we would use 'van' : Lucas van Leiden ( from the city of Leiden ), Kees van Hattem, Tom van Amsterdam etc. I don't think having any such name adds to your social status. In the Netherlands a double surname has a certain status as it suggests having anchestors belonging to nobility, e.g. J.W. Wolff van Deken.

The short answer is yes, it is both a French preposition and a fixed part of the family name. "De" means "of" so "Stael" would (usually) have been the place the family was originally from.