The newest release will always be found at github.com/openlilylib/LO-ly/releases here on GitHub. Look for an *.oxt file with a name similar to OOoLilyPond-0.X.X.oxt and download it:
For anyone who doesn't want to read the release notes, there's a simple Download page as well.
In LibreOffice, open the extension manager (Tools -> Extension Manager), click the "Add" button which will open a file dialog. Select the *.oxt file you've just downloaded and confirm with the "Open" button.
When asked for whom you want to install the extension, you can choose "only for me" which won't require administrator privileges on your system. After successful installing, close the extension manager, and probably you will be requested to restart LibreOffice.
Now LibreOffice will have a new "OLy" toolbar. It contains a single "OLy" button that launches the extension.
By the way: Please make sure that you have LilyPond installed on your Computer. ;-)
Congratulations - you are now ready for the next chapter: Introduction.