Installation 

This version supports TYPO3 14.3 LTS with PHP 8.2 to 8.5. For TYPO3 12.4 and 13.4 LTS use version 13.x of the extension, for TYPO3 11.5 LTS version 12.x.

Composer 

composer require svent/pageclasses
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Then update the database schema, e.g. in System > Maintenance > Analyze Database Structure.

Without Composer 

Install the extension pageclasses from the TYPO3 Extension Repository (TER) in System > Extensions. The database fields are created during the installation.

Include the TypoScript 

Add the static template Pagewide CSS Classes (pageclasses) to the main TypoScript template of your site: Sites > TypoScript, edit the whole TypoScript record and add it in the field Include static (from extensions).

That's all: the TypoScript of the extension sets page.bodyTagCObject (see setBodyTagCObject), so the classes are rendered without further configuration.