How does the Ring doorbell get powered?
tl;dr version: Ring wires across the doorbell switch pair and sources power from the bell transformer. 

The switch wires provide Ring with low-current 16~24V AC through the bell coil. Ring's current load is low enough that it doesn't trigger the bell. This small current is (usually) enough for Ring to keep up the charge its local battery. More about ‘usually’ below.

Ring's power management works as follows. Most of the time, Ring is in a deep sleep mode and uses very little power - a trickle for the battery, and a small current for the passive IR sensors and a small low-power MCU. During deep sleep, the camera and radio are off and the MCU is monitoring the array of IR sensors on the front. When the MCU senses motion on the IR, it wakes up the rest of the system and starts recording / transmitting the potential porch pirate's misdeeds.

If you're lucky enough to have a legit visitor instead of a porch pirate, they will eventually press the doorbell button on the Ring (unless they're millennial, in which case they will text you. Whatever.) The button press makes Ring short across its power feed to ring the bell, cutting off its power. But no worries: Ring uses its battery to power itself even while the doorbell switch is pressed, so you can see your visitor staring alternately at their phone and their shoes.

And... the ‘usually’. The new Ring Pro version seems to use more power, more than can be reliably sourced through the bell coil in some cases because of the coil DC resistance. The Power Kit you show is designed  to shunt some additional current past the doorbell coil to supply the Ring Pro connected at the door. It seems complex - would be interesting to analyze it.