When the required checkbox option is enabled the field will have to have a value before the record can be fully saved in the record editors on the web app or the mobile apps.
On the web app this means that the field will need a value before you can save the record.
On the mobile apps (Android, iOS), if records can be saved as a draft (allowable by default), then the user will be presented with an option to save the record in a draft state if any required fields do not have values. When records are saved as a draft they only exist on the mobile app and are not synced to the server until completed.
If you are importing data, the importer does not enforce requirement logic and records will be created/edited without the value in the fields. The next time the record is edited after it has been imported the fields that are required will need to have values.