I download from the Libre website, open the .dmg file. copy the icon over to application but do not any installation feedback, only that it has been copied.
On OSX, for software distributed as disk images (.dmg), installation is as simple as copying the .dmg file to applications. Does LO now appear in your Applications folder? Does LO appear in Launchpad? You should be able to start it from either location.
If you have trouble starting LO and see messages about ‘verifying’ or ‘unknown developers’, have a look here: Safely open apps on your Mac – Apple Support (AU) or change your settings under System Preferences >> Security and Privacy.