What function does the stage two regulator play while diving?

This second part of the regulator helps manage pressure that comes from the first stage via the hose, air that is then delivered to the mouthpiece through which you breathe.  Unlike the first stage, the second stage is more complicated in that it process air inhalation and exhalation.  With the valve being opened by the piston or diaphragm, when you inhale through the mouthpiece, air pressure within the chamber is reduced.  From there, water pressure pushes in the diaphragm, opening the intake valve.  After you stop breathing in, air pressure within the chamber balances, causing the valve to close.  This process is what allows air you breath but without air leaking from the mouthpiece during the process.