In the Checklist in Case of Problems page there is bullet point stating:
Make sure this version of Java is selected in LibreOffice (Tools -> Options -> Advanced) or OpenOffice.org (Tools -> Options -> Java)
Two bullet points above this one there is one stating:
For Mac users: LanguageTool requires Java 8 or later, which is available for the Mac only as a 64 bit version. You need the 64-bit version of LibreOffice 4.2 or later in order to use LanguageTool on a Mac and you also need to have 'Java for OS X' installed, additionally to Java 8 (Source: #74877).
I am using LibreOffice 188.8.131.52 and it is working, but the instructions were a bit confusing. I am not sure, but it seems that I am not using Java 1.6 (at least not pointing to it on Preferences/Options -> LibreOffice -> Advanced) and it still works.
If need be, I will be happy to help change the text on the page. Just let me know how.