Here’s the scenario:
You’re running at WOT? in 3rd gear and your turbo is pushing as much air as it can.
Your RPM approaches the redline.
You press the clutch and lift your foot off the gas to shift into 4th.
The throttle closes.
The air coming out of the turbo has nowhere to go.
This is where the blow-off valve comes into action - by venting, it keeps the intake pressure from spiking, and allows the turbocharger to keep spinning and keep moving air. Thus when you open the throttle again, the turbo is still spinning at full speed (or close to it) and the intake pressure is reasonable, and you when you open the throttle again, the engine is still able to make plenty of power.
Note that if the MAF is of the draw-through? type, there is a risk that the car will run rich when the BOV opens, since the ECU is providing fuel in proportion to the measured airflow, and some of that air isn’t actually making it to the engine. A bypass valve, or BPV, solves this problem by recirculating the vented air into the intake path, upstream of the turbo. This way the MAF sees only as much air as is actually entering the engine, so the ECU can maintain the desired AFR.
A BOV can be quite loud; a hiss can often be heard clearly when BOV-equipped cars are shifting or decelerating.
TODO: describe advantages of BOV over BPV.