The answer for Linux Users ;)
- Download the LanguageTool for desktop
- Unpack the downloaded .zip file (just provisionally) in your download folder. The terminal command is, for example,
unzip LanguageTool-stable.zip
- In the unpacked folder you will find the version subfolder
LanguageTool-5.5
(of course the exact name depends on the LanguageTool version)
- Move the version subfolder to an appropriate directory. I personally recommend the root directory
/opt
. Therefore, write in terminal sudo mv LanguageTool-5.5 /opt
- Adapt the Settings in Texstudio (
Options > Configure TeXstudio > Language Checking > LanguageTool
) There you introduce following: (see picture)
a. The path of the languagetool-server.jar
file in the LT Path field. In our case we type: /opt/LanguageTool-5.5/languagetool-server.jar
b. The path to the java platform in the Java field. In our case we type: /usr/bin/java
(of course it is prerequisite that you hava java installed)

Thanks to @shivang-patel for providing the updated link to the application. And to @soham for giving the new name of the .jar
file.
In old versions of LanguageTool (e.g., version 4.7
) the name of the file is languagetool.jar