1. Make sure you have downloaded the Linux version of the game
2. Once you have downloaded it, extract the archive by rightclicking it and choosing "extract here" from the context menu
3. Go into the extracted game's folder and doubleclick on the file "Aftertaste.sh". If it asks you whether you want to display the file's contents or run the file, you click on "Run"
4. Now the game should start!
I am personally running Linux Mint, so the UI may vary but this is how it should work