As the silver is disolved into the water the resistance of the water continuously falls, and the current wants to continually rise. The LM334Z however does not allow the current to rise. Instead it steadily lowers (regulates) the voltage, so the current remains constant.
V = IR, so if R (resistance) is falling, then I (current flow) must increase if V (voltage) is constant.
Regarrange this to read I = V/R
To hold I constant while R is steadily falling, V must also steadily fall. The LM334Z accomplishes the task of decreasing V to hold I at a constant value quite well.