The solenoid B1033 opens the spring-loaded valve B1032 to control the timing of suction for the desoldering gun. Specifications indicate that the controller delays 0.2 sec after the button is depressed before suction is applied, and again delays 0.5 sec after the button is released before terminating suction. One reason you would want to use a solenoid to control suction, instead of relying on starting/stopping the pump, is to prevent a pressure pulse from pushing air outward. It also helps to prevent the tip from clogging.
The solenoid most likely operates from the main DC voltage which is 24V I think? I have a similar unit I can check.