The iCloud syncing switch in Mental Case is disabled when iCloud is not available to the app. There can be two reasons for this:
1) You are not logged into an iCloud account in the Settings
app
2) You are logged into iCloud, but you have not turned on syncing
of 'Documents & Data' in the list of things to put in
iCloud.
If you make sure you are logged in, and Documents & Data is on, then when you relaunch Mental Case, you should be able to switch on iCloud syncing.