What does the metric condition $\nabla_\rho g_{\mu\nu}=0$ in General Relativity intuitively mean for an observer measuring distances?

The condition $\nabla_{a}g_{bc} = 0$ is just pure mathematics. Every metric admits a torsion-free (for one defintition, one that satisfies $\nabla_{[a}\nabla_{b]}f = 0$ for every function on the manifold) connection that satisfies this condition.

That general relativity is formulated using this connection is a statement that gravity obeys the equivalence principle -- a freely falling observer is parallel translated along the geodesics of $\nabla$ relative to $g$. And the fact that this is a parallel translation is encoded in that condition in the metric.