Here is another explanation with more options...
The console is sending weather data to some weather software device/system on the local network. This could be WeeWx or Weather-Display or a Meteobridge...etc. Then that device has a template populating service running in this case W34's template and it sends the weather data to a personal website on the Internet. That personal website is just a normal web sever that has been configured to receive the template data. There are several of these weather templates available; W34, Meteotemplate, Saratoga, and Leuven are a few popular ones also, and there are others. These templates run on many different weather software applications. Where you host your personal website doesn't really matter. Most people buy a domain name and set up hosting with some web hosting company that they like. You can even host your own web server if your ISP allows the port 80 web browser port, does require static IP or dynamic DNS service, or you can put your web server on a non-standard port that only you know for personal use.