Sometimes there is a problem with the install process, and the game isn't able to write the save files. All you need to do is delete the game from your iPad (hold the icon until it becomes wiggly, then tap the "X" and confirm that you want to delete it).

After that, tap the Home button, then open the App Store and reinstall Fish Tycoon (you won't be charged for it, even if you see a message asking if you want to purchase it). Let the game install, and the game should save your progress.