As you click any world in play menu, another menu will be opened: Play, Favourite, Delete (if yours).
When you click play, you appear in a new world, loaded specifically for you <3
In all games you should be presented with RESET button, which reacts differently on each DNW game, more about it on each game page. Also you get Menu button, which opens the same world specific menu as you got in play menu, except the Play button is replaced by Restart, but it does the same thing - loads up a world for you.
Favouring a world, or faving, is like giving likes on social media. This, unlike many social sites will save the world in your favourite list, which you can access in play menu. Also favouring world will allow their authors to get bigger planes (reminding: 100 favs for 4x4, 200 for 6x6) for more interesting adventures!