RingPatrol connects to the low voltage side of the transformer, which is generally between 8 and 24 VAC. This voltage is low enough that it is safe to touch without removing power. This is why national electrical codes allow the transformer terminals to be exposed, unlike high voltage (120VAC) wiring that must be in a plastic or metal junction box.
To be safe, RingPatrol recommends that you either verify that the voltage with a voltage meter (to be absolutely sure the transformer hasn't failed) or remove power at your breaker box. It is an absolute requirement to remove power if you have a pacemaker or any other sensitivities to low voltage.
The challenge is finding which breaker powers the doorbell. The location of the transformer installation is a big clue. Look at the labels on the circuits and find the one that best matches the location of the transformer.
Note: Please ensure when you are removing power that the other devices powered by that circuit will not be affected by removing power. For example, if your transformer is attached to your furnace, please ensure you turn off your thermostat, and your furnace is not running before you remove power.