The Prius C uses a key-less ignition system. That means you'll use a press button to turn the car on and off, instead of turning a key.
You can find the button to the right of the steering wheel, under the right side air vent - there's a photo below.
Note: It won't work unless you have the brake pedal down. So put your foot on the brake, press the button, and you're good to go!
Only the electric engine will kick in right away, so don't worry if you don't hear anything. If you can see the dash light up, you're set.