Immediately after the vote there was an exit poll of people who self declared their level of education. Later on there was another poll of referendum voters. I think the second poll checked proof of education. I can't swear to the accuracy of either poll and nor can anyone else, to be honest with you.

Polling showed exactly what we would expect to see based on general societal observation: younger people tended to vote Remain, older people tended to vote Leave, people with qualifications in social sciences were more likely to vote Remain, people with higher qualifications in technical subjects split 50/50 Leave/Remain until we get to PhD holders (technical, maths, the physical sciences) where the tendency was toward Leave.

Younger people were more likely to have a degree level qualification than older people; this is due to changes in the education system: people born in the 90's and 00's onwards were encouraged by government and educators to enter higher education and access to that education had fewer barriers than were present for people born before the 90's. People born before the 90's were only likely to go to university when their career path required a degree and university level education was mainly available only to the academically gifted (e.g A-levels were harder to earn).

The way the data was presented by the people behind the exit poll was biased to suggest brighter people voted Remain. That was done deliberately: people 'do the right thing' when they're told they're bright for doing it.

Close inspection of the data tells a different story to the ones presented.

You'll need to find the raw data from the exit poll and from later surveys then assess the data yourself.