For Mountain Lion, verify that you downloaded LibreOffice v5.4.5.
You double-click the application installer, and then drag the application image within the installer onto the link to the applications folder, also in that installer panel. Then eject the installer from your Desktop.
Next, since this application did not come from the Mac App Store, you have to verify it with Apple's Gatekeeper. In the Applications folder, right-click on the LibreOffice icon, and choose Open from the pop-up dialog. You will see a verification progress bar, and you may have to select Open again to actually open the application. This is a one-time verification ritual, and then you can double-click on the LibreOffice icon to launch it normally.
Also, LibreOffice may pop a dialog about needing Java. Disregard. In the LibreOffice Preferences : Advanced panel, ensure that the use a Java Runtime environment is unchecked. This will gag future dialogs on this subject. You won't need Java unless you are doing database activity in LibreOffice.