I’m not sure it actually reduces the time of the gear shifts themselves. Well, it shouldn’t.
Basically it should reduce transient lag. Which is the time it takes a turbo to get back on boost when the throttle body opens.
So for example a car with a huge turbo. At full throttle all the way through the revs from say 2000RPM it starts making boost at 4500RPM, hits full boost of 25PSI at 5000RPM, which falls off to 20PSI at the limiter which is at 7500RPM.
Say you’re gunning it full throttle in 3rd gear, change up a gear at 7300RPM: the throttle closes as you change up, much less exhaust gasses are flowing through the exhaust manifold with a closed throttle so the turbo loses boost, back onto the throttle at 5500RPM in 4th, the turbo then needs to build boost again as exhaust gases build flow again through the exhaust manifold.
This is transient lag. Different to the lag you experience at low-mid revs with an open throttle (where there’s simply not enough exhaust gas being blown out by the engine at low-mid revs to spool the turbo in the first place). The lag at low revs under an open throttle cannot be improved by an anti-lag system. But transient lag, the small delay in boost that occurs getting back on the throttle in a rev range where the turbo would ordinarily make healthy boost, can be improved by an anti lag system.
There are a few ways of doing anti lag, but generally it involves creating combustion in the exhaust manifold under a closed throttle (where there would usually be much less gas flow under a closed throttle). Provoking combustion in the manifold under a closed throttle creates gas flow which spools the turbo.
The advantage is a reduction in transient lag. The car won’t be quicker under open throttle and established boost. It doesn’t increase horsepower. It just makes the turbo react much faster, when in normal circumstances it would take more time to reach full boost again after a gear change, or a quick lift of the right foot before going full throttle again.