That's not true. While most voters from low income households supported leaving the EU (as shown in the report you linked), 59% of leave votes came from middle class voters and only 17% of Leave votes were from skilled manual workers. The majority of middle class voters supported remaining, but because there are so many of them, middle class voters also made up the majority of leave voters. This 5 minute report [0] from the BBC looks into where the votes for Brexit came from and how the different classes voted.
[0] https://www.youtube.com/watch?v=eOMiUONDLno