There could be lots of reasons.
Students coming from state schools could have been a big fish in a small pond, and have to get past a crisis of confidence before they excel. Perhaps a 3 year degree doesn't give them enough time to reach that point.
If other top universities don't see the same pattern if private school students getting more firsts than state school students, then it could be the assessment process in Cambridge that makes things different, degree classification gained solely from terminal exams. Private school pupils are given at least annual, and often twice yearly exams in exam conditions. Perhaps these habits make the difference.
Private school pupils also have a different relationship with their teachers and become more accustomed to asking for and receiving support. Perhaps this carries on throughout university.
How many of the genius firsts came from private school students who had scholarships or bursaries and so had already been recognised and taken into the private school system, so now represent firsts from a private school, rather than a state school. In which case, we should be calling for more selection at age 11 or 13 and bringing back assisted places.
What does the data look like between STEM and none STEM? Is this marked difference for firsts between state and private students seen more in humanities and social sciences?
It sounds like an interesting puzzle to solve. Ultimately though, what does it mean? If a state school pupil gains a place at Cambridge and doesn't get a first, does it affect how capable they are in the various employment sectors or entrepreneurship?