How does intranet work without using an actual internet connection?

The same way an internet connection is transmitted through a network: cables and wireless.

Strictly speaking, internet connection isn’t transmitted through a network. It is something a network has access to, through said cables and wireless technologies. When you connect to WiFi, you’re not connecting to the internet. You are connecting your device to another device which itself has access to the internet.

All of this is to say that networks work perfectly fine without internet connections. Ethernet and wireless technologies are not inherently for the internet. They are technologies which facilitate the running of networks in general.

Servers and clients, likewise, do not require the internet. If you have a server which is a local server (i.e. a local media server) then your device can communicate with that server using whatever protocols it has to use provided the devices are connected somehow.