LanguageTool can be used in a different number of ways:
Double click the downloaded LanguageTool-3.x.oxt to install it. If that doesn't work, call Tools -> Extension Manager -> Add.... Close LibreOffice/OpenOffice.org and re-start it. Type some text with an error that LanguageTool can detect and you should see a blue underline. You might want to use "This is an test." as an example – make sure the text language is set to English for this example.
Download the *.zip version (not the *.oxt one) and unzip it. Then start languagetool.jar by double clicking it. If your computer isn't configured to start *.jar archives, start it from the command line using java -jar languagetool.jar.
You can use the --tray option to start LanguageTool inside the system tray. After you copy any text to the clipboard, clicking LanguageTool in the system tray will cause the application to open and check the contents of the clipboard automatically. This way you can use LanguageTool for applications that do not support direct integration of the checker.
See "as a stand-alone application" above, but start LanguageTool using:
java -jar languagetool-commandline.jar <filename>
Use java -jar languagetool-commandline.jar --help to get detailed usage information.
A list of all command-line options is also available in our Wiki.