The installation of the extension in your shop works like any other Shopware 6 extension and is thus done in a few steps.
Extensions > Store and search for "Google Tag Manager". Click on the extensions and then on the button
Add extension to download it.
Go to the installation in the admin under
Extensions > My extensions > Apps and click on the button
... and then click on
In the same place as during the installation, you can now access the admin under
Extensions > My extensions > Apps to change the status to