the wastegate is held closed due to a hard spring, pre-tensioned to only allow opening of the gate after a certain boost can force the spring enough to allow the wastegate to open. this movement on the wastegate happens via a daprhragm in the actuator/external wastegate actuator which opposses the spring fed from the compressor side of the turbo.
you can get more boost by allowing a bit of the air going to the actuator device to "bleed" off, therefore reuding the effective pressure forcing the spring back, therefore holding the wastegate shut longer and allowing more boost.

the ecu or in some cases an electronic standalone controller will be causing the bleeding to be metered to attain certain boost levels.
in crude setups a mechanical bleed valve is used and that allows a fixed amount of air to be bled off with no user control in the car, and can be prone to failing and big boost occuring momenterily causing damage.

in all cases there needs to be an actuator of sorts to keep tension holding gate shut, that can move quickly enough, most mechanical solenoids would burn out holding enough tension on a wastegate for a long time i would have thought.

p.s. wrote this a bit sleepy, just a guide on how it all works and feel free to correct and fundamental errors.