there's a meaningful payload to orbit advantage with hot staging that is conservatively about a 10% increase
(Elon Musk during Twitter Spaces interview on June 24th, 2023, timestamp 37:13)
In a line of work dominated by tiny incremental sub-cent percentage improvements, this ten percent figure is quite a lot. I can understand why there would be an improvement, I just don't understand how it's so much.
My understanding is that after separation, the second stage is in free-fall and is not experiencing the acceleration necessary to pull the liquid fuel towards the bottom of the spacecraft. In order to push the liquid to the bottom, either an ullage motor is used to provide some small acceleration (with the propellant being some kind of pressurised gas I assume) or a rubberised bag pressurised with helium is used in the fuel tanks themselves to push the fuel aftwards. Hot staging is another way of solving this problem without using the above methods, since the upper stage is still experiencing acceleration.
With hot staging, you get the mass saving of not having any ullage motors or such, but you need extra mass on the first stage to resist the blast from the second stage engines. You get to fire the second stage immediately rather than wait a few seconds for the first stage to drift clear. Again, I see how this could be an improvement, but I don't see how it could add up to so much.