Kierkegaard did use a technique called "indirect communication" through which he sought to emphasize the need for the reader to actually engage what was being said. In his case, this is as specific response to Danish Hegelianism which claimed that everything was understood. For Kierkegaard, the point is that some things (specifically, things like what it would mean for God to become a man, the resurrection) were, in fact, not being understood at all but rather presumed to be understood.
There are some 20th and 21st century "continental" philosophers who write in unclear ways. They might claim that their practice has an origin in Kierkegaard, but I don't know of any who do so for the same motive. Maybe in the broadest terms they do, because they might deny that everything is understood or that language can help us to understand everything. Surely, however, they don't agree with Kierkegaard's specific claims. In a weird way, they also see the specific claims of Christianity as speed bumps on the way to something else.
In regards to who consciously engages in this practice among 20/21st century continental figures, it would be nearly impossible to catalogue them all. Derrida, at least according to my dissertation advisor, promised his mother never to write clearly.
Regarding who we can confidently state does this, I don't know how that would be anything but contentious. First, the true believers in whichever figure X may deny that it's obscurantist. Second, there is a legitimate distinction between really hard to understand and gibberish.
So for instance, returning to Kierkegaard, I wouldn't say every single sentence in every single work is lucid prose that is easy to decipher. But I do think he's actually a very good stylist, and the difficulty in understanding his philosophy stems not from obscurantism but from the wide-ranging source materials he's familiar with and the fluency he has in using them (incorporating deep knowledge of standard interpretations of the Greek and Latin philosophers).
Sharing just my own opinions on some of the bigger figures:
Husserl - very difficult ideas, very bad writing, not with the goal of obscuring
Heidegger - simpler ideas, bad readings of other texts, not as hard to understand if you can read German, sometimes obscuring in style due to idiosyncratic usages. Sometimes obscurant for philosophical reasons related to his views of our understanding.
Sartre - clearer if you understand Hegel and Husserl, difficult due to the style, not prone to obscurantism
Levinas - hard to follow because of idiosyncratic vocabulary. In desperate need of an editor, somewhat intentionally obscurant due to opposition to what he sees as traditional modes of relating to the world
Foucault - hard to follow to due to historiographic method of argumentation
Derrida - sometimes intentionally obscurant
Those are the main continental figures I know from the 20th and 21st century.
In regards to "when did this practice start," without a clearer idea of what the practice is, I have no idea.
In regards to "what would be their exact reasons," this part of your question is fundamentally unanswerable. Because (1) there may not be a common practice, (2) each thinker may have their own reasons for doing similar or common practices, and (3) no thinker was forced to write down why they wrote in the way we did.
If you become a philosopher, I would strongly recommend against adopting the practice. While you might find a few universities that will let you write in this way, my experience has been that regardless of sub-field, philosophical writing is best done with clarity and directness. Or to put it another way, academic philosophy is different from famous people philosophy.
The so-called "analytic continental" divide is a weird one, because "analytic" at this point refers to a method (having shifted from the claim that denied there is an metaphysics and saw philosophy just as the analysis of language) -- seeking clarity in our writing, and "continental" refers to a subject matter, i.e. certain philosophies from the continent of Europe starting from about Hegel onward.