It sounds like you're trying to have the SDL framework copied into your game's application bundle when you build the project, but the framework isn't being copied. Since you don't have the SDL framework installed on your system, the game doesn't work because there's no SDL framework on the system. You can check to see if the SDL framework is in your game's application bundle by selecting the game in the Finder, right-clicking, and choosing Show Package Contents. The SDL framework should be under Contents > Frameworks in the application bundle.
If the SDL framework isn't in the application, you have to tell Xcode to copy it. Instructions for doing this are in the following article:
Read the comment from Adam on Feb. 2, 2012 in the article. He has a workaround if the SDL framework isn't being copied to the application bundle.