Some of the software is free.
You still need a server that is reachable over the internet by dns and runs that software. Which for commerical games means renting server capacity and registering a domain name. It's a similar topic as how to run your own website.
If you have a server that runs on one of the player's machines, you need that player to overcome the router issue. That's not that hard, but it's a hassle for casual players. You need some port forwarding and use ddns or rely on finding out the external adress and giving those to the other players.
Maybe there is a free service somewhere for connecting machines to play together. But I guess that's hard to find. Multiplayer games are usually commercial or have local multiplayer. I quick search came up with some adware services and I do not know how trustworthy those are. Also it seemed like they were running the game, not merely connecting the machines.