[Edmund Burke] was not a man of the enlightenmentThe idea here (clearer with context) is that the violent, radical, utopian French Revolution, and associated thought, is the True Enlightenment, and all the other attempts at progress don't get to count as part of The Enlightenment.
So, if you're an avid reformer, a man of reason and thought, but also a man of non-violence who wants to move forward with sufficient error correction rather than without it -- as Burke was -- then you're a stodgy old unEnlightened conservative.
According to Amazon reviews, the book is biased to the right wing and gives a very favorable treatment of Burke. Those Amazon reviews must have double the left-wing bias that Kirk has. Equal bias would make them see it as fair, and then they need to go left again to see it as being slanted right.