Signing in with Game Center will save your Kingdom Rush 6: Genesis progress automatically. This means you'll be able to pick up where you left off, even if you uninstall the game or switch phones, as long as you use the same account and connect to the cloud.
You can check that your cloud save is enabled by following these steps:
Go to Settings > scroll down and tap iCloud > under "Recommended for you", tap the option "See all" > search for Genesis and scroll the button to the right to enable cloud save if it's turned off.