When you click submit the data is submitted as part of the HTTP request to the web server. If the form is a GET type then the data values are sent as part of the URL. If the form is a POST type then it is encoded and sent after the HTTP request headers.
What the web server does with the form values is completely configurable by the web server administrator/programmer.
If your web server supports CGI (common gateway interface) and a CGI script is setup then the POST values will be decoded, stored in the environment, the launched CGI script will be able to read the values from the environment, and store them on the filesystem or the database.
Of course if your using PHP or the more advanced server side technologies for handling forms, it skips the CGI step, and the web server will communicate the transmitted values directly to the server side technology.