In a Riemannian manifold such as the surface of a sphere you could travel a distance $\epsilon$ in every direction and note that you get a different amount of surface area or volume than if you did so in a flat space. For instance if you walked a distance equal to the distance from the north pole and the south pole ($2\pi R$) you get an area of $4\pi R^2,$ whereas if you walked that same distance in a flat space you get $\pi (2\pi R)^2$ which is larger.
For a Lorentzian manifold such as a spacetime it is a trace of the Riemann curvature tensor. Both of these things and the fact that they are different are mentioned in the wikipedia page you cite.
edit to respond to edited question
The full quote is
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold
Emphasis added.
Given we have a metric describing spacetime,
Not a Riemannian metric, just a nondegenerate symmetric bilinear form with a Lorentzian signature. What in GR you call a metric, but not what a mathematician would think you meant if you just said the word metric out of context, so not necessarily what they are talking about if they just say the word metric.
(b) the volume of the ball does not differ from that of a Euclidean ball when $R_{\mu\nu}=0$?
There is no obvious ball in spacetime, since intervals can be positive or negative. So what would be the ball about some event? Would it include every event that is lightlike separated, every timelike separated event, every spacelike one? One of those and then some more too?
If your goal is to understand the meaning of the Einstein Equation, there are better sources.