This sentence uses 不屑于, not 不屑, but the meaning would be something like:
Also, some leaders of “小资女人” started to look down on speaking English, and they sometimes think that to speak Japanese or French makes it clear that they are different from others.
The translation might be a bit awkward so feel free to edit it.
I feel that the 于 here takes an object (说英文) after that. But in other usages, 不屑 just takes an object directly, without any preposition.
Or if you relate it to English, it looks like 不屑 here works as an intransitive verb, and 于 is a preposition. But in other uses I have seen, it works as if a transitive verb.
However, why does 不屑 need to take 于 here, even though in other cases it does not need?