How do I keep / restore iOS 6 app icons on my iPhone?
Apple gave iOS a complete overhaul with the release of iOS 7 in 2013. Unfortunately, there is really no way to avoid the change, unless you stick to iOS 6. This means you won't be able to download any of Apple's future updates if you want to keep the current look of your device.
The only other possibility is jailbreaking your phone. As of right now, you can't jailbreak beyond iOS 6.1.3. This means you will have to wait until someone jailbreaks iOS 7, and hope that some kind of iOS 6 theme hits Cydia.
UPDATE: You can finally change your iOS 7 icons back to iOS 6 icons with the new jailbreak theme "iOS 6 Icons For iOS 7 Theme". The theme is available in the BigBoss repo via Cydia, and it requires Winterboard to install. You can learn how to jailbreak your iPhone or iPad running iOS 7 using Windows here and Max OS X here.