The water pressure pressing against a diaphragm, plus additional pressure from a spring, act to operate a valve. When the demand-side pressure is too low the valve opens. Pressure on the supply side pushes water into the demand side, raising its pressure. As the demand-side pressure increases the diaphragm moves and the valve gradually closes. Eventually the demand-side pressure rises to a level that results in the diaphragm closing the valve fully and no additional water (and pressure) from the high side is allowed to pass.
There is a training video which gives an animated explanation of the process in a Watts brand regulator at Youtube there.
If the regulator setting is turned down while there is no flow then one of two things must happen: either the demand-side pressure remains as-is until an outlet is opened, or else the regulator itself provides a leakage path to bleed off the excess pressure. The latter is common with air pressure regulators because usually leaking off a bit of air won't make a mess. The former is common with water pressure regulators because they're often installed in a location where leakage would be problematic.