Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci square, Kretschmann and Weyl square?
In $n$ dimensions, the Ricci scalar curvature $R$ at a point measures how much the volume $V$ of a small $n$-dimensional ball of radius $\epsilon$ around that point differs from the Euclidean value $V_E$:
I am not aware of similar formulas for higher-order invariants, but that doesn’t mean they don’t exist.
Since all of the invariants are built from the Riemann curvature tensor, and the curvature of a manifold is entirely a geometric concept, it is somewhat odd to be asking for geometrical descriptions, but I assume you are looking for something simpler than the obvious description in terms of the curvature tensor.