If you are wanting to transfer your game progress and accomplishments from one iPhone, iPad or iPod Touch to another so that you don't have to start over again, or if you'd like to share your progress with friends, iExplorer is the app to do it.
Apple did make this much more difficult and limited starting in iOS 8.3 and later by blocking some app folder access, so if you do have iOS 8.3 or later, you may only be able to transfer the game saves to your computer. The article below explains more:
App folders in iOS 8.3 are empty. Why is that?
Game save files are usually located in the Documents folder of an app, though each app places the save file(s) in a slightly different location and with slightly different names. The Library subfolder is also sometimes used, so you may want to contact the developer of the app to know specifically where to look and to find out the specific file paths.
Before attempting to transfer, it's recommended to make sure the app is not running in the background on your iPad, iPhone, or iPod Touch when manipulating files like this. To make sure the app is not running in the background, you can double tap the Home button and find the app in the recents list. Then, in iOS 7 or 8, swipe up on the app window to fully close it, or in iOS 6 and under, hold your finger on the icon of the app until a red minus sign appears and tap on the red minus sign.
It's also a good idea to back up each device in iTunes before trying this, just in case you make a mistake and need to restore.
TRANSFER TO COMPUTER (BACKUP YOUR GAME PROGRESS)
Once you find the right file path(s), just select the file(s) in the iExplorer window (hold Control (PC) or Command (Mac) to select more than one) and drag/drop the file to your Mac or PC. Or, you can right-click and choose Export to Folder.
NOTE: If you have iOS 8.3 or later and don't see the app folder or files you need, you may need to go into the Backup Explorer section instead (within the Backups section). Please see this article for more information and how to do it: App folders in iOS 8.3 are empty. Why is that?.
TRANSFER TO DEVICE
This has been made much more difficult to do by Apple starting in iOS 8.3, so you may not be able to transfer game save or other app data to your device if you have iOS 8.3 or later. The article at the bottom of the paragraph directly above above explains more about it.
If the game does have iTunes File Sharing and the data does get stored in the Documents folder or you have a Jailbroken device, to then add the game progress to an iPhone, iPad, or iPod Touch, you'll want to add the file(s) to the same location on the destination device and choose to replace the existing file(s). You can either drag/drop into the device directory or right-click in the iExplorer window and choose Add Files.
After replacing the file(s), relaunch the app on your iPhone, iPad, or iPod, and you should see the progress has been transferred. If not, you should try resetting the device (like restarting a computer) and check again. To do this, simply press and hold the Home button and the sleep/wake button (on the top) at the same time for around 10 seconds until you see the white Apple logo appear.
If you are copying the entire Documents or Library folders to the top level of an app folder and choosing Replace, and that's not working, instead, you'll want to open the Documents or Library folders and copy over the folder contents (the files within those folders) and replace the existing files in those folders.
These instructions will work for most apps in the App Store. The only apps you won't be able to transfer games saves by using iExplorer are the apps that use Game Center to track game progress. These are usually the ones that require an Internet connection to play like most of the freemium apps with in-app purchases, so the only way to load your game progress onto a different device for those apps is to log into Game Center on that device. The other exception is for apps on devices running iOS 8.3 or later that don't have iTunes File Sharing enabled or keep the save files in the Library folder which can't be accessed currently (iOS 8.3 only).
NOTE: The Documents and Library subfolders of an app will in most cases have read and write privileges, but the .app (dot app) subfolder of an app is read-only in iOS 6 and 7 and is not even accessible in iOS 8 or 9 unless you Jailbreak your device. The article below explains more:
Why can I no longer make changes to the .app (dot app) subdirectory of an app?