What are these electrical boxes in the closet that have no breakers in a house with buttons for the light switches?
Most likely the boxes contain relays that are controlled by low-voltage circuits from the push buttons.   A relay is an electrical switch that is controlled by an electrical circuit.   In this case, the relay allows low-voltage wiring and push buttons to control the light circuit.

This setup was sometimes used before modern home automation systems existed to allow control of a circuit from multiple locations or with devices that were not rated for normal line voltages, and a variation of it is sometimes used today where the switches are located near the circuit panel and remote switches are located in convenient locations.