Double click the home button to reveal the multitasking tray, and swipe from left to right on it, so you see the player controls (on the iPhone, you have to swipe 2 times). Tap the AirPlay icon and pick your Apple TV.
If your device supports mirroring, you'll see a toggle that lets you enable it - do that, and you should see your device screen appear on your TV.
If you don't wan't the interface elements to show on the TV when you're inside ShutterSnitch, you can go to the Settings app and find ShutterSnitch in the Apps section. Scroll down to the bottom, and you'll see an "External screen" option that you can set to "Fullscreen viewer" instead of "Mirrored output".