Going to 400Hz rather than say 50/60Hz does not save much copper weight but it saves the iron which makes up the magnetic components of transformers and motors.
In a typical 50Hz 3000VA transformer (that uses reasonable quality laminations) the core would weigh around 20kg.
At 400Hz this goes down to around 5kg; a big weight saving.
There is some copper weight saving because the turns per volt scale inversely with the cross sectional area of the core etc and a smaller transformer will need a shorter length of wire to make each turn around the core. But there is no wire diameter saving because the current is still the same.
Skin effect is negligible at 400Hz.
The other reason for 400Hz is that in the old days one did have solid state DC-AC inverters; one used a motor driving a generator to do such conversions. And the output frequency is the RPM divided by the # of poles of the rotor. The smallest number of poles physically possible is 2 and that means a 50Hz generator has to run at 3000rpm. This is inconveniently slow. A 400Hz generator can run at 24000rpm or, more usefully, can have more poles for the same rpm and thus be more compact.
Nowadays one would not bother with 400Hz because of solid state inverters using lightweight ferrite cores and operating at tens of kHz, but this is old history.