I can't see any Apple TV content to browse when I open the Apple TV app. could you help me with this?

Your Apple ID account has a country setting that determines the region for Apple services, as well as valid payment methods. VPN has nothing to do with it, as Apple doesn’t use IP addresses for allowing or blocking access.

What do you see instead in the Apple TV app?

The GUI of the TV app may differ a bit per macOS version, so please confirm Monterey if that applies.

Watch Apple TV+ content on a Mac - Apple Support

Do you see an Apple TV+ option in a side bar on the left, or along the top of the window?

Note that Apple TV+ subscribers need to additionally sign in to the TV app (menu Account) on Mac, in order to get playable access to subscription content.

The TV app would show only the Library when it has no internet connectivity to load more online content. When the TV app detects [functional] internet access, then it should show more services.

Else the country or region could restrict what is supported in the TV app, as Apple doesn’t bring video services to some select countries at this time. Availability of Apple Media Services (by country) - Apple Support.