Reference: here
Given 3-dim Euclidean metric in spherical coordinates by
$$ds^2 = dr^2 + r^2 d\theta^2 + r^2 \text{sin}^2 \theta d\phi^2 \tag{1}$$
so restricting to
$$r=R=const. \tag{2}$$
where $R$ is scalar Ricci, gives
$$ds^2 = R^2 d\theta^2 + R^2 \text{sin}^2 \theta d\phi^2 \tag{3}$$
Questions:
What is the physical meaning of $r=R=const.$ as stated in $(2)$?
What is the relation between $r$ and $R$?
The metric you present uses curvilinear coordinates for flat (non-curved) R$^3$ (Euclidean 3-d space).
When you restrict to r = constant, the resulting spherical surface is 2-dimensional and curved.
For any curved space (or space-time) you can go through often lengthy calculations to compute the Ricci scalar, R. For a spherical surface, R happens to have the same value as the radius of the sphere, r.
On a 2-d surface, the scalar R tells us how much a gyroscope (vector that keeps pointing "in one direction" turns when you move in a small loop. In higher dimensions, the tensor R$^a_{bcd}$ hold more information, as loops in different directions and differently pointing vectors, end up turning by different amounts. R does then not have have this simple interpretation, but still plays a crucial role in Einstein's equations, relating curvature to mass.
