If you don't want the savegames to be actual files, then the easiest method would be to use localstorage. It allows your website to save several MB of key-value pairs in the user's web browser. The main advantages are that this is completely invisible to the user and does not require a server. The main problem with this is that savegame data is bound to the web browser. When the user wants to play on multiple devices, they can not easily transfer savegames.
Another option is to set up a server with a web API which stores the game states in a database. This will also require a user account system. The game you used as a template seems to use the Dropbox API for this.