The Tesla app, like many (most?) mobile apps, is always running. The only time it isn't running is if you explicitly force it to close (in developer options > running services on Android). Restarting the phone will start the service again though.
This is likely the same on iOS devices. You don't need to "open" the Tesla app for phone as key to function. All you need to do is install the app, login to your account, and setup phone as key while in the car.
Then you "use" phone as key by just walking up to your car and getting in, and finally driving away. As you approach the car, the car and phone will Bluetooth pair and the car will then unlock when you pull the door handle as well as "start" so that you can drive without a key card.