Phone Disk works by allowing you to mount your iPad, iPhone or iPod touch on your Mac or PC just like it was an attached drive. By default, Phone Disk gives you access to the music, video and photo media stored on your iOS device (root-level access to the OS itself still requires a jailbreak). However, by selecting "Change Connection Root" in the Phone Disk menu bar item, you can gain access to the file system for any of the third-party apps you've installed. If you've ever right-clicked on one of your Mac's applications and chosen "Show Package Contents," this is essentially the same principle; most Mac OS X and iOS applications are really a package of individual files bundled together in a folder that gives the appearance of a single file.
The mess that Mac OS X and iOS normally hide from you
Gaining access to these bundled files in your third-party iOS apps via Phone Disk is how you can locate game saves for the apps in question. Beyond this step, however, you're going to need a bit of ingenuity to figure out exactly which file contains your save data. Part of the reason Apple doesn't provide easy syncing of game saves between iOS devices may be because app developers haven't adopted a standardized method of encoding this information within their apps.
To give you an idea what I'm talking about, here are a few examples. The game saves for Square-Enix's iOS app Chaos Rings are stored in the app's Documents folder, and the save file itself is called data.dat. Game saves for Infinity Blade are also stored in the app's Documents folder, but this time the file is called SwordSave.bin -- not just a different name, but also a different file format. Angry Birds is even messier, with several files in its Documents folder storing data related to your game progress. Pride and Prejudice and Zombies stores its save information in a .plist file buried within the app's Library/Preferences folder.
Because of this scattershot approach to game save info, even with Phone Disk granting you access to apps' internal file structure, transferring save data from one device to another isn't a particularly user-friendly process. Fortunately, Phone Disk allows you to mount multiple iOS devices at once, allowing you to compare and contrast the file directories between devices. If you see a file that's present on your first device (the one with save data) but not present on the second device (the one with a fresh install of the app), chances are pretty good that file has settings, saves or other info that you'll want to transfer over. All you need to do is drag the file from one device to another, making sure the file ends up in the same location on the second device.
One of these things is not like the other
To give you an idea of how the process works, here's the procedure I followed to transfer my Infinity Blade saves from an iPhone 4 to an iPad 2. First, I connected my iPhone 4 to my Mac and let it sync with iTunes as normal (ensuring I had all my data backed up). Then, I used Phone Disk to mount the iPhone 4 on my Mac, then selected "Change Connection Root" to mount the Infinity Blade app. Infinity Blade's save file is easy to find; it's at the first level of the app's Documents folder, and it's conveniently named SwordSave.bin.
Phone Disk's menubar item makes Finder access to apps simple
Next, I connected my iPad 2, let it sync to iTunes to get its data backed up, and mounted it onto my Mac in Phone Disk. I selected "Change Connection Root" to the Infinity Blade app, then navigated to the app's Documents folder. And what do you know, there was no SwordSave.bin file there. Going back to the Finder window with my iPhone 4's copy of Infinity Blade showing, I dragged SwordSave.bin from that window to the iPad 2's window, which copied the file over. I unmounted the iPad 2, disconnected it from my Mac and fired up Infinity Blade on my iPad 2.
Voila, Infinity Blade launched on my iPad 2 in exactly the same spot I'd left it on my iPhone 4. Every last bit of my save data was intact, from what weapons I'd bought, how much gold and XP I had, even to what point in the castle I was last.
The above procedure is going to be different for each game since, as mentioned, iOS developers haven't settled on a standard model for storing game saves yet (nor are they necessarily ever going to, until/unless Apple provides a default framework and format for game saves). The key here is not being afraid to experiment. As long as you don't modify anything on your original device, you're at no risk of losing your game saves. Even if something gets botched in the transfer to the second device, you can always delete and resync the app on the second device and try again. The process isn't something I'd recommend you vouch to your 8-year-old for transferring Angry Birds saves, but if you've got a moderate level of expertise tooling around in Mac OS X's Finder (or Windows Explorer), you should be able to figure out the rest on your own.
It's also worth mentioning that Phone Disk is not the only utility out there that'll give you access to your iOS apps in the Mac OS X Finder. There are many apps that offer the same functionality; the only reason I've focused on Phone Disk is because that's what I used. (Fellow TUAWer TJ Luoma notes that PhoneView from Ecamm Networks can be used to achieve the same ends.) Also, if you've already jailbroken your iDevices, you've got root-level access to their file systems anyway, so a solution like Phone Disk isn't for you. But if you're not jailbreaking and don't want to, either Phone Disk or an app like it will give you just enough access to your iDevice's files to let you work around some of the iOS platform's restrictions.