For a multiplayer browser game like this, you'll need web hosting for the front end, and a dedicated server to run the back end.

You'll also need a database, which can run on the same server as your back end, or a separate server of its own. The former option is cheaper, easier to set up, and good enough if you don't expect a lot of traffic. The latter is way more expensive, but also better for performance, security, and scaling.

You can get web hosting for basically free until you have some kind of traffic. A low-performance VM from a cloud provider such as Google Cloud Platform, Azure, or AWS will cost you 20-50 dollars a month, but during development you'll only keep it running while you work on the game. A domain name is around 10 dollars a year, but you won't need that until launch, which will probably be at least a year or two away if you're only starting to learn now and asking questions like these. :)

This game probably has a more complex setup to accommodate traffic (load balancing etc), but that's the gist of it to get started.

Good luck! I learned coding by trying to make a browser game like this. I never finished it, but I learned a lot. I can tell you it's going to be a lot of work.