I came across this question, the answer achieves what it does, but there is no explanation of how it works. I have some ideas that {} serves as a placeholder for \newtheoremstyle. I tried to experiment a bit and added one more {} to the command \newtheoremstyle{case}{}{}{}{}{}{:}{ }{}, but this failed.
In another comment of the question, {} was also used in \newenvironment, for example, \newenvironment{}{}{}.
I could not find any reference as to what {} actually does in the context of Latex in general. Where can I find some references of the command {}?
{}are empty mandatory arguments to the command\newtheoremstyle. See theamsthmmanual for the meaning of all the arguments. And because the number of mandatory arguments of\newtheoremstyleis fixated it also does not make sense to add one more{}. Moreover, this question looks like a XY Problem. So: Please, ask the real problem and illustrate it by a minimal working example.{}do in the command\newenvironment{}{}{}?\newenvironment{}{}{}would define a new environment without name, without begin code, and without end code. And because this would redefine\end, it would break LaTeX. So it would not only not make any sense, but also would not be a good idea. Please read a LaTeX introduction or the answer already referred in the comment to the question you've linked to for more information about\newenvironmentand other basics.