The template game is a game with only one room, and all the menus, keys etc. already set up for you, along with names for most variables used and comments here and there. It is recommended you choose this option, as you can jump straight in and start programming your first room from here. For more information on the template game, see the readme.txt file in the template directory.
When you create a game, only the data files (and source code, if based on the template game) are placed in the directory you chose. There is no interpreter there. To run the game, you will need to install an AGI interpreter and specify the interpreter command line in the "Settings" menu. You don't have to install it in the same directory with your game - enough that it will be accessible by PATH, or just specify the full path in the "Settings". The interpreter must be version 2.915 or higher (but not version 3). You can usually find out the version number of an interpreter when you enter debug mode in a game.
At the time of writing, none of the Linux AGI interpreters are fully functional. Sarien is the most advanced but unfortunately, the Sarien's development had stopped. You can download Sarien from http://mega-tokyo.com/sarien/ or http://agi.helllabs.org.
If you want to test every function of your game, you'll have to run it in MSDOS. It is quite convenient to run it under DOSEMU (in console mode - the XDOSEMU currently doesn't support 16-color mode). In that case you'll have to copy the interpreter from another game or even from an AGI demo. Apparently Sierra still doesn't allow to distribute their interpreter, so it is not included together with the template game.
See the files used by AGI section to see what files you need to copy into the game directory. If you copy the files into the template directory, then whenever you create a new game from the template these will be copied also.
Note: If you are using a newly written AGI interpreter, then the version number will probably be different but new interpreters should support this format. The files you have to copy over will also be different.