To your specific questions:

why people use traefik

Reverse proxies let you serve multiple apps from one server, encrypt your traffic and automate updates of encryption certificates. I use Apache (virtual hosts, mod_md) for this.

how and why to use docker containers

Modern web apps have many constantly shifting dependencies and don't work closely with distributions on packaging them, so they have to be updated separately from the base system. They might also need different versions of these dependencies, and multiple instances of one dependency (e.g. database). Containers are one way to achieve that, and also ease backups, monitoring of individual apps, and installing an application in exact same way on development and testing machines.

what settings to change on the router

To expose ports 22 (ssh), 80 (http) and 443 (https) to world, forward them to your server.

how to connect and setup the domain

In the DNS settings for your domain, point "A" record to your home IP. That's if you rent a static IP. If you don't, things will be more complicated, research on your own.

how to do this safely

Bare minimum: set up daily backups of container volumes and reverse proxy configuration, same for updates of base system and containers, and turn off SSH password login in favor of keys. Test that a backup can be successfully restored in a virtual machine at least once a month.