Interpreting the Kretschmann scalar

For vacuum solutions, since the Ricci tensor $R_{ab}$ vanishes, the Kretschmann scalar is equal to the norm of the Weyl tensor, $K = C_{abcd}C^{abcd}$. This means it is telling you something about the tidal forces at a given point. I might use $K^{1/2}$ to characterize the strength of the tidal forces. This can be used in Schwarzschild or Kerr spacetimes to see that the tidal forces go like $M/r^3$ (at least in the equatorial plane for Kerr).

The Kretschmann scalar can be used as an indicator of curvature singularities in the manifold. For instance, in the Schwarzschild black hole (given in the Wikipedia link in your post), $$ K\propto\frac1{r^6} $$ so as $r\to0$, $K\to\infty$.