are you running on LINUX or Windows?
if you are on Windows, double-clicking on the icon should be enough to start up ENIGMA.
If you are using LINUX and you double-click on the lateral GM icon, it will only startup the IDE without the compiler, which seems to your case. You should have a file named 'Start.sh' in your 'ENIGMA-DEV' directory. This you need to make executable by right-click once, then left-click 'Properties'. From there you go to the 'Permissions' tab, and make sure the 'run as Executable' check box is ticked.
Afterwards you double click and run the 'start.sh' script and when LGM should now have the green button present.