Hi kiwiandpeticot! I would first confirm that you have the Mac version of the game (the second download option). It should show up on your computer as a file named the-pirate-mermaid-demo-osx.zip. If I remember correctly, all you need to do is double click that zip file, and the game application will pop out (it will have an icon with the main character's face on it). Double click the icon/application to start the game.
If that doesn't work, let me know. It's been a while since I've tried the game on a Mac, so I might need to check later on another computer. ^^