A lot of replies, but no one bothers to actually try and fix the issue OP is having. Not that Lutris and minigalaxy won't work around his problem (they probably will), but running .sh files on Solus should work fine.
mojave21
First step: make sure to go to the file's properties (right-click -> properties -> permission tab) and set the permission to execute ("Allow executing file as program"). Most likely it does not run because this is not set.
This is working as designed to prevent users executing unwanted software (e.g. malware) on the system (I think).
When you made sure this is set, run the file. If you still get no installer window popping up, run it from the terminal. You don't need sudo in most cases, so just run it like this:
$ ./gog_trine_enchanted_edition_2.0.0.2.sh
Please post here what the terminal states after you run it. That would give the clue about why it does not run (permission, library missing, etc.).
