I believe those use a hydraulic accumulator system rather than compressed air.

Guess what's on the other side of the membrane or piston in a hydraulic accumulator?

I would guess that it would be a metal spring of some sort that doesn't lose energy to thermal side effects of compressing and decompressing gas, for a higher round trip power efficiency. Or else it could be a relatively large gas reservoir so that it doesn't heat or cool the gas too much.

I'm far from foiling boats would you mind explaining why are these compressors human powered and not gas/battery powered? Is it something from racing code?

Racing code - America's Cup races prohibit stored energy aboard the boat, for example. Exceptions are made for certain springs, valves, and batteries for electronics.