Understanding kurtogram parameters
[Given figures are from A new improved Kurtogram and its application to planetary gearbox degradation feature analysis, Xianglong Ni et al.]
This $k$ in the graph (and the $K$  in the Table) corresponds to an equivalent &quot;dyadic-power&quot;, providing the number of subsequences at each level, alternating dyadic splits with filters on subbands $[0,1/4]/[1/4,1/2]$ and $[0,1/6]/[1/6,1/3]/[1/3,1/2]$.   The number of such subsequences in the 1/3-binary tree are $2,3,4,6,8,12\ldots$, approximately $2^1,2^{1.6},2^{2},2^{2.6},2^3,2^{3.6}\ldots$, since $\log(2)/\log(3)\sim 0.6$.
Additional sources:

Calcul Rapide du Kurtogramme et Applications, Jérôme Antoni (short, in french)
Fast computation of the kurtogram for the detection of transient faults, Mechanical Systems and Signal Processing, 2007

I unfortunately cannot help totally on the interpretation of this particular paper. I remember that the original  spectral kurtogram was meant to detect, localize or characterize nonstationarities from signals. The kurtogram was a fast version, reusing principles from wavelet packets and multiband filter banks and quasi-analytic filters.  When looking at maximum values in a kurtogram, if they are concentrated, one may suspect non-stationary components in a noisy signal. Then, it is possible to extract or invert the aforementioned component to check or denoise the original data.
I would suggest you to read several papers, especially the early ones, showing interpretations of different kurtograms.