Here is a link to a diagram and description of the operation.
The spring which is adjusted by the adjusting screw acts against a diaphragm that is pressurized by the output pressure. If the pressure is too low, the spring force acts on a valve mechanism to open it, admitting flow from the high pressure source. If the pressure it too high the force acts against the spring so the valve will not be opened.
Different regulators may have different details but they act on the same principle.
Increasing the flow often causes the output pressure to be lower because there are pressure losses in the mechanism. The loss of pressure is sometimes called "droop". Regulators are often rated by "droop" versus flow rated.
Some regulators use a pilot mechanism (more comples) that permits greater flow with less droop.