The idea is to grow a hypersphere around each point and estimate the dimensionality of its neighborhood within that sphere. then you take the average of the dimensionality over all points.
i.e. with two points in the neighborhood, the dimensionality is 2, but if the next added point when growing the neighborhood is almost colinear, it’s still not much bigger than 2D. and so on.
then you record the growth of dimensionality per unit of sphere growth, and find a/the maximum.